id	question	solution	final_answer	context	image	modality	difficulty	is_multiple_answer	unit	answer_type	error	question_type	subfield	subject	language
0	"如图, 在等腰 $\triangle A B C$ 中, $A B=B C, I$ 为内心, $M$ 为 $B I$ 的中点, $P$ 为边 $A C$ 上一点, 满足 $A P=3 P C, P I$ 延长线上一点 $H$ 满足 $M H \perp P H, Q$ 为 $\triangle A B C$ 的外接圆上劣弧 $A B$ 的中点. 证明: $B H \perp Q H$.

<image_1>"	['取 $A C$ 的中点 $N$, 链接 $IN, BQ, QC$, 如下图. \n\n<img_4177>\n\n由 $A P=3 P C$, 可知 $P$ 为 $N C$ 的中点.易知 $B, I, N$ 共线, $\\angle I N C=90^{\\circ}$.由 $I$ 为 $\\triangle A B C$ 的内心, 可知 $C I$ 经过点 $Q$, 且\n\n$$\n\\angle Q I B=\\angle I B C+\\angle I C B=\\angle A B I+\\angle A C Q=\\angle A B I+\\angle A B Q=\\angle Q B I,\n$$\n\n又 $M$ 为 $B I$ 的中点, 所以 $Q M \\perp B I$, 进而 $Q M \\| C N$.\n\n考虑 $\\triangle H M Q$ 和 $\\triangle H I B$. 由于 $M H \\perp P H$, 故\n\n$$\n\\angle H M Q=90^{\\circ}-\\angle H M I=\\angle H I B .\n$$\n\n又 $\\angle I H M=\\angle I N P=90^{\\circ}$, 故 $\\frac{H M}{H I}=\\frac{N P}{N I}$, 于是\n\n$$\n\\frac{H M}{H I}=\\frac{N P}{N I}=\\frac{1}{2} \\cdot \\frac{N C}{N I}=\\frac{1}{2} \\cdot \\frac{M Q}{M I}=\\frac{M Q}{I B}\n$$\n\n所以 $\\triangle H M Q \\sim \\triangle H I B$, 得 $\\angle H Q M=\\angle H B I$. 从而 $H, M, B, Q$ 四点共圆, 于是有\n\n$$\n\\angle B H Q=\\angle B M Q=90^{\\circ} \\text {, }\n$$\n\n即 $B H \\perp Q H$.']			['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proof	Plane Geometry	Math	Chinese
1	"如图, $A, B, C, D, E$ 是圆 $\Omega$ 上顺次的五点, 满足 $A B C=B C D=C D E$, 点 $P, Q$ 分别在线段 $A D, B E$ 上, 且 $P$ 在线段 $C Q$ 上. 证明: $\angle P A Q=\angle P E Q$.

<image_1>"	['设 $A D$ 与 $B E$ 交于点 $S, C Q$ 的延长线交圆 $\\Omega$ 于 $T$, 如下图.\n\n<img_4200>\n\n注意到 $A B C=B C D=C D E$, 因此 $A B, C D$ 所对的圆周角相等, 设为 $\\alpha, B C, D E$所对的圆周角相等, 设为 $\\beta$. 于是\n\n$$\n\\begin{aligned}\n& \\angle A T Q=\\angle A T C=\\alpha+\\beta, \\\\\n& \\angle P T E=\\angle C T E=\\alpha+\\beta, \\\\\n& \\angle P S Q=\\angle B D A+\\angle D B E=\\alpha+\\beta .\n\\end{aligned}\n$$\n\n由 $\\angle A T Q=\\angle P S Q$ 可得 $S, A, T, Q$ 四点共圆, 由 $\\angle P T E=\\angle P S Q$ 可得 $P, S, T, E$ 四点共圆. 所以\n\n$$\n\\angle P A Q=\\angle P T S=\\angle P E Q\n$$']			['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']	Multimodal	Competition	False				Theorem proof	Plane Geometry	Math	Chinese
2	"如图, 点 $A, B, C, D, E$ 在一条直线上顺次排列, 满足 $B C=C D=\sqrt{A B \cdot D E}$, 点 $P$在该直线外, 满足 $P B=P D$. 点 $K, L$ 分别在线段 $P B, P D$ 上, 满足 $K C$ 平分 $\angle B K E$, $L C$ 平分 $\angle A L D$.

证明: $A, K, L, E$ 四点共圆.

<image_1>"	['令 $A B=1, B C=C D=t(>0)$, 由条件知 $D E=t^{2}$.\n\n注意到 $\\angle B K E<\\angle A B K=\\angle P D E<180^{\\circ}-\\angle D E K$, 可在 $C B$ 延长线上取一点 $A^{\\prime}$,使得 $\\angle A^{\\prime} K E=\\angle A B K=\\angle A^{\\prime} B K$, 并连接 $LE$, 如下图.\n\n<img_4165>\n\n此时有 $\\triangle A^{\\prime} B K \\sim \\triangle A^{\\prime} K E$, 故 $\\frac{A^{\\prime} B}{A^{\\prime} K}=\\frac{A^{\\prime} K}{A^{\\prime} E}=\\frac{B K}{K E}$.\n\n又 $K C$ 平分 $\\angle B K E$, 故 $\\frac{B K}{K E}=\\frac{B C}{C E}=\\frac{1}{t+t^{2}}=\\frac{1}{1+t}$. 于是有\n\n$\\frac{A^{\\prime} B}{A^{\\prime} E}=\\frac{A^{\\prime} B}{A^{\\prime} K} \\cdot \\frac{A^{\\prime} K}{A^{\\prime} E}=\\left(\\frac{B K}{K E}\\right)^{2}=\\frac{1}{1+2 t+t^{2}}=\\frac{A B}{A E}$.\n\n由上式两端减 1 , 得 $\\frac{B E}{A^{\\prime} E}=\\frac{B E}{A E}$, 从而 $A^{\\prime}=A$. 因此 $\\angle A K E=\\angle A^{\\prime} K E=\\angle A B K$.\n\n同理可得 $\\angle A L E=\\angle E D L$.\n\n而 $\\angle A B K=\\angle E D L$, 所以 $\\angle A K E=\\angle A L E$. 因此 $A, K, L, E$ 四点共圆.']			['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proof	Plane Geometry	Math	Chinese
3	"如图, $\triangle A B C$ 为锐角三角形, $A B<A C, M$ 为 $B C$ 边的中点, 点 $D$ 和 $E$ 分别为 $\triangle A B C$ 的外接圆弧 $B A C$ 和弧 $B C$ 的中点, $F$ 为 $\triangle A B C$ 的内切圆在 $A B$ 边上的切点, $G$ 为 $A E$ 与 $B C$ 的交点, $N$ 在线段 $E F$ 上, 满足 $N B \perp A B$.

证明: 若 $B N=E M$, 则 $D F \perp F G$. 

<image_1>"	['作辅助线如下图.\n\n<img_4139>\n\n\n由条件知, $D E$ 为 $\\triangle A B C$ 外接圆的直径, $D E \\perp B C$ 于 $M, A E \\perp A D$. 记 $I$ 为 $\\triangle A B C$的内心, 则 $I$ 在 $A E$ 上, $I F \\perp A B$. 由 $N B \\perp A B$ 可知 $\\angle N B E=\\angle A B E-\\angle A B N=$ $\\left(180^{\\circ}-\\angle A D E\\right)-90^{\\circ}=90^{\\circ}-\\angle A D E=\\angle M E I$. (1)\n\n又根据内心的性质, 有 $\\angle E B I=\\angle E B C+\\angle C B I=\\angle E A C+\\angle A B I=\\angle E A B+$ $\\angle A B I=\\angle E I B$, 从而 $B E=E I$. 结合 $B N=E M$ 及 (1) 知, $\\triangle N B E \\cong \\triangle M E I$. 于是 $\\angle E M I=\\angle B N E=90^{\\circ}+\\angle B F E=180^{\\circ}-\\angle E F I$, 故 $E, F, I, M$ 四点共圆. 进而可知 $\\angle A F M=90^{\\circ}+\\angle I F M=90^{\\circ}+\\angle I E M=\\angle A G M$, 从而 $A, F, G, M$ 四点共圆.再由 $\\angle D A G=\\angle D M G=90^{\\circ}$ 知, $A, G, M, D$ 四点共圆, 所以 $A, F, G, M, D$ 五点共圆.从而 $\\angle D F G=\\angle D A G=90^{\\circ}$ ，即 $D F \\perp F G$.']			['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proof	Plane Geometry	Math	Chinese
4	"如图所示, 在平面直角坐标系 $x O y$ 中, $A, B$ 与 $C, D$ 分别是椭圆 $\Gamma: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>$ $b>0)$ 的左右顶点与上下顶点. 设 $P, Q$ 是 $\Gamma$ 上且位于第一象限的两点, 满足 $O Q \| A P$, $M$ 是线段 $A P$ 的中点, 射线 $O M$ 与椭圆交于点 $R$.

证明: 线段 $O Q, O R, B C$ 能构成一个直角三角形.

<image_1>"	['设点 $P$ 坐标为 $\\left(x_{0}, y_{0}\\right)$. 由于 $\\overrightarrow{O Q}\\|\\overrightarrow{A P}, \\overrightarrow{A P}=\\overrightarrow{O P}-\\overrightarrow{O A} ; \\overrightarrow{O R}\\| \\overrightarrow{O M}, \\overrightarrow{O M}=\\frac{1}{2}(\\overrightarrow{O P}+$ $\\overrightarrow{O A})$, 故存在实数 $\\lambda, \\mu$, 使得 $\\overrightarrow{O Q}=\\lambda(\\overrightarrow{O P}-\\overrightarrow{O A}), \\overrightarrow{O R}=\\mu(\\overrightarrow{O P}+\\overrightarrow{O A})$. 此时点 $Q, R$的坐标可分别表示是 $\\left(\\lambda\\left(x_{0}+a\\right), \\lambda y_{0}\\right),\\left(\\mu\\left(x_{0}-a\\right), \\mu y_{0}\\right)$. 由于点 $Q, R$ 都在椭圆上, 所以 $\\lambda^{2}\\left(\\frac{\\left(x_{0}+a\\right)^{2}}{a^{2}}+\\frac{y_{0}^{2}}{b^{2}}\\right)=\\mu^{2}\\left(\\frac{\\left(x_{0}-a\\right)^{2}}{a^{2}}+\\frac{y_{0}^{2}}{b^{2}}\\right)=1$.\n\n结合 $\\frac{x_{0}^{2}}{a^{2}}+\\frac{y_{0}^{2}}{b^{2}}=1$ 知, 上式可化为 $\\lambda^{2}\\left(2+\\frac{2 x_{0}}{a}\\right)=\\mu^{2}\\left(2-\\frac{2 x_{0}}{a}\\right)=1$, 解得 $\\lambda^{2}=$ $\\frac{a}{2\\left(a+x_{0}\\right)}, \\mu^{2}=\\frac{a}{2\\left(a-x_{0}\\right)}$.\n\n\n\n因此 $|O Q|^{2}+|O R|^{2}=\\lambda^{2}\\left(\\left(x_{0}+a\\right)^{2}+y_{0}^{2}\\right)+\\mu^{2}\\left(\\left(x_{0}-a\\right)^{2}+y_{0}^{2}\\right)$\n\n$=\\frac{a}{2\\left(a+x_{0}\\right)}\\left(\\left(x_{0}+a\\right)^{2}+y_{0}^{2}\\right)+\\frac{a}{2\\left(a-x_{0}\\right)}\\left(\\left(x_{0}-a\\right)^{2}+y_{0}^{2}\\right)$\n\n$=\\frac{a\\left(a+x_{0}\\right)}{2}+\\frac{a y_{0}^{2}}{2\\left(a+x_{0}\\right)}+\\frac{a\\left(a-x_{0}\\right)}{2}+\\frac{a y_{0}^{2}}{2\\left(a-x_{0}\\right)}$\n\n$=a^{2}+\\frac{a y_{0}^{2}}{2}\\left(\\frac{1}{a+x_{0}}+\\frac{1}{a-x_{0}}\\right)$\n\n$=a^{2}+\\frac{a y_{0}^{2}}{2} \\cdot \\frac{2 a}{a^{2}-x_{0}^{2}}=a^{2}+\\frac{a^{2} \\cdot b^{2}\\left(1-\\frac{x_{0}^{2}}{a^{2}}\\right)}{a^{2}-x_{0}^{2}}=a^{2}+b^{2}=|B C|^{2}$.\n\n从而线段 $O Q, O R, B C$ 能构成一个直角三角形.']			['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']	Multimodal	Competition	False				Theorem proof	Plane Geometry	Math	Chinese
5	"如图所示, 在等腰 $\triangle A B C$ 中, $A B=A C$, 边 $A C$ 上一点 $D$ 及 $B C$ 延长线上一点 $E$满足 $\frac{A D}{D C}=\frac{B C}{2 C E}$, 以 $A B$ 为直径的圆 $w$ 与线段 $D E$ 交于一点 $F$. 证明: $B, C, F, D$ 四点共圆.
<image_1>"	['如下图, 取 $B C$ 中点 $H$, 则由 $A B=A C$ 知 $A H \\perp B C$, 故 $H$ 在圆 $w$ 上.\n\n<img_4161>\n\n延长 $F D$ 至 $G$, 使得 $A G \\| B C$, 结合已知条件得, $\\frac{A G}{C E}=\\frac{A D}{D C}=\\frac{B C}{2 C E}$, 故 $A G=$ $\\frac{1}{2} B C=B H=H C$, 从而 $A G B H$ 为矩形, $A G H C$ 为平行四边形. 由 $A G B H$ 为矩形知, $G$ 亦在圆 $w$ 上. 故 $\\angle H G F=\\angle H B F$. 又 $A G H C$ 为平行四边形, 由 $A C \\| G H$,得 $\\angle C D F=\\angle H G F$. 所以 $\\angle C D F=\\angle H B F=\\angle C B F$, 故 $B, C, F, D$ 四点共圆.']			['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proof	Plane Geometry	Math	Chinese
6	"如图, 在 $\triangle A B C$ 中, $A B=A C, I$ 为 $\triangle A B C$ 的内心. 以 $A$ 为圆心, $A B$ 为半径作圆 $\Gamma_{1}$, 以 $I$ 为圆心, $I B$ 为半径作圆 $\Gamma_{2}$, 过点 $B, I$ 的圆 $\Gamma_{3}$ 与 $\Gamma_{1}, \Gamma_{2}$ 分别交于点 $P, Q$ (不同于点 $B$ ). 设 $I P$ 与 $B Q$ 交于点 $R$. 证明: $B R \perp C R$.

<image_1>"	['如下图, 连接 $I B, I C, I Q, P B, P C$.\n\n<img_4195>\n\n由于点 $Q$ 在圆 $\\Gamma_{2}$ 上, 故 $I B=I Q$, 所以\n\n$$\n\\angle I B Q=\\angle I Q B .\n$$\n\n又 $B, I, P, Q$ 四点共圆, 所以 $\\angle I Q B=\\angle I P B$, 于是 $\\angle I B Q=\\angle I P B$, 故 $\\triangle I B P$ 与\n$\\triangle I R B$ 相似, 从而有 $\\angle I R B=\\angle I B P$, 且\n\n$$\n\\frac{I B}{I R}=\\frac{I P}{I B}\n$$\n\n注意到 $A B=A C$, 且 $I$ 为 $\\triangle A B C$ 的内心, 故 $I B=I C$, 所以\n\n$$\n\\frac{I C}{I R}=\\frac{I P}{I C}\n$$\n\n于是 $\\triangle I C P$ 与 $\\triangle I R C$ 相似, 故\n\n$$\n\\angle I R C=\\angle I C P .\n$$\n\n又点 $P$ 在圆 $\\Gamma_{1}$ 的弧 $B C$ 上, 故\n\n$$\n\\angle B P C=180^{\\circ}-\\frac{1}{2} \\angle A,\n$$\n\n因此\n\n$$\n\\begin{aligned}\n\\angle B R C & =\\angle I R B+\\angle I R C \\\\\n& =\\angle I B P+\\angle I C P \\\\\n& =360^{\\circ}-\\left(90^{\\circ}+\\frac{1}{2} \\angle A\\right)-\\left(180^{\\circ}-\\frac{1}{2} \\angle A\\right) \\\\\n& =90^{\\circ},\n\\end{aligned}\n$$\n\n故 $B R \\perp C R$.']			['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proof	Plane Geometry	Math	Chinese
7	"如图, 点 $D$ 是锐角 $\triangle A B C$ 的外接圆 $\omega$ 上弧 $B C$ 的中点, 直线 $D A$ 与圆 $\omega$ 过点 $B, C$的切线分别相交于点 $P, Q, B Q$ 与 $A C$ 的交点为 $X, C P$ 与 $A B$ 的交点为 $Y, B Q$ 与 $C P$ 的交点为 $T$. 求证: $A T$ 平分线段 $X Y$.



<image_1>"	['首先证明 $Y X \\| B C$, 即证 $\\frac{A X}{X C}=\\frac{A Y}{Y B}$. 连接 $B D, C D$, 如下图.\n\n<img_4225>\n\n为了方便我们直接用三角形代表该三角形的面积, 因为\n\n$$\n\\frac{\\triangle A C Q}{\\triangle A B C} \\cdot \\frac{\\triangle A B C}{\\triangle A B P}=\\frac{\\triangle A C Q}{\\triangle A B P}\n$$\n\n于是\n\n$$\n\\frac{A C \\cdot C Q \\cdot \\sin \\angle A C Q}{A B \\cdot B C \\cdot \\sin \\angle A B C} \\cdot \\frac{A C \\cdot B C \\cdot \\sin \\angle A C B}{A B \\cdot B P \\cdot \\sin \\angle A B P}=\\frac{A C \\cdot A Q \\cdot \\angle C A Q}{A B \\cdot A P \\cdot \\sin \\angle B A P}\n$$\n\n根据弦切角定理, 有\n\n$$\n\\angle A C Q=\\angle A B C, \\angle A C B=\\angle A B P\n$$\n\n又 $D$ 是弧 $B C$ 的中点, 于是\n\n$$\n\\angle C A Q=\\angle D B C=\\angle D C B=\\angle B A P,\n$$\n\n于是可得\n\n$$\n\\frac{A B \\cdot A Q}{A C \\cdot A P}=\\frac{C Q}{B P}\n$$\n\n因为 $\\angle C A Q=\\angle B A P$, 所以 $\\angle B A Q=\\angle C A P$, 于是\n\n$$\n\\frac{\\triangle A B Q}{\\triangle A C P}=\\frac{A B \\cdot A Q \\cdot \\sin \\angle B A Q}{A C \\cdot A P \\cdot \\sin \\angle C A P}=\\frac{A B \\cdot A Q}{A C \\cdot A P}\n$$\n\n\n\n而\n\n$$\n\\frac{\\triangle B C Q}{\\triangle B C P}=\\frac{B C \\cdot C Q \\cdot \\sin \\angle B C Q}{B C \\cdot B P \\cdot \\sin \\angle C B P}=\\frac{C Q}{B P}\n$$\n\n因此可得\n\n$$\n\\frac{\\triangle A B Q}{\\triangle A C P}=\\frac{\\triangle C B Q}{\\triangle B C P}\n$$\n\n即\n\n$$\n\\frac{\\triangle A B Q}{\\triangle C B Q}=\\frac{\\triangle A C P}{\\triangle B C P}\n$$\n\n又\n\n$$\n\\frac{\\triangle A B Q}{\\triangle C B Q}=\\frac{A X}{X C}, \\frac{\\triangle A C P}{\\triangle B C P}=\\frac{A Y}{Y B}\n$$\n\n故 $\\frac{A X}{X C}=\\frac{A Y}{Y B}$.\n\n设 $B C$ 边的中点为 $M$, 因为\n\n$$\n\\frac{A X}{X C} \\cdot \\frac{C M}{M B} \\cdot \\frac{B Y}{Y A}=1\n$$\n\n所以由塞瓦定理的逆定理可知 $A M, B X, C Y$ 三线共点于 $T$, 故由 $Y X \\| B C$ 可知 $A T$平分线段 $X Y$.']			['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oAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAGsuQen0NG08c59zTqKAGhcCgAjqcmnUUAFFFFABRRRQA0qTnpz6iuOvfBU+mXcup+DrxNKvJCXlsXXdZXLY/ijH3GOB86YPHINdnSYoA4608eQ2l0un+K7NtBvicJLO+bWfrzHN07fdbDcjiuwV1ZQykFSMgjkEVDd2Vvf2strdwRXFvKNskUqBlYehB4NcofAs+kM0vhLWbjSed32GYG4s27keWxymfVGGPSgDstwpa4n/AISrxDo2V8R+GJ5Y1wDe6KxuYz2yYziRR9A1bGkeNPDeuyCLTtZtZZ84MDP5coPvG2GH5UAb1FJmgEEZoAWik3CjNAC0UZpMigBaKM01pERSzMAo5JPQfWgB1JkVy998Q/DVpO1rBfHUr0A4tNMja6lJHUYQEKf94iqh1Hxpr3Gm6Xb6BaN/y86mfOuMdisKHaD/ALzH6UAdVf6lZaVZyXmoXUNrbRjLyzOEUfia5A+KtY8TjZ4P0/baNnOs6ihSIDHWKPh5Dg8EhV9zVyy+H+nLeJqGtXFxr2pLys+okOkZ/wCmcQGxBnkYGfeur24AA6DtQBzug+DbPSLttTuZ5tU1qRcSajeYMmO6oBxGvJ+Vce+a6MClAwMUUAHaiiigApMUtFAFO90nT9SCi/sLW6C9PPhV8fTIqS2sreziENrBFBCvSOJAqj8BxViigA7UUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFACEVl6v4a0TXk26rpVneYGFaaIMy/Ruo/A1q0UAccPh1ZWe3+xtZ1zSAv3Y7a/d4v+/cu9f0pV0PxtZljbeL7S8X+FdQ0tSfxaJ0/wDQa7CigDjlb4iw8NB4XugB95Zp4D19Crjp71I+p+OkwF8NaNL6lNXcfzhrrDwKO2aAOT/tDx5LHlfD+hRHPSTVZG/lDUYj+Is7cy+F7NMn7qz3BA4x/wA8x612A6Z70tAHIDw14qunJ1DxvNHGRjy9O0+KHH/An8xv5UR/Dbw/I6S6r9u1uZCSH1W8knH/AHwTs/8AHa6+igCpY6bZaZbi3sLO3tYB0jgiVFH4AVaxilooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooA//Z']	Multimodal	Competition	False				Theorem proof	Plane Geometry	Math	Chinese
8	"如图所示, 在 $\triangle A B C$ 中, $X, Y$ 是直线 $B C$ 上两点 ( $X, B, C, Y$ 顺次排列), 使得

$$
B X \cdot A C=C Y \cdot A B
$$

设 $\triangle A C X, \triangle A B Y$ 的外心分别为 $O_{1}, O_{2}$, 直线 $O_{1} O_{2}$ 与 $A B, A C$ 分别交于点 $U, V$. 证明: $\triangle A U V$ 是等腰三角形.

<image_1>"	"['如下图, 设圆 $O_{1}$ 与圆 $O_{2}$ 的公共弦为 $A D, A D$ 交 $X Y$ 于 $E$.\n\n\n\n<img_4145>\n\n由于 $A D$ 为两圆的根轴, 于是 $E$ 点对圆 $O_{1}$ 和圆 $O_{2}$ 的幂相等, 从而\n\n$$\nX E \\cdot C E=B E \\cdot Y E\n$$\n\n进而结合合分比定理有\n\n$$\n\\frac{B E}{C E}=\\frac{X E}{Y E}=\\frac{X B}{Y C}\n$$\n\n又由已知, 有 $\\frac{X B}{Y C}=\\frac{A B}{A C}$, 于是有 $\\frac{B E}{C E}=\\frac{A B}{A C}$, 从而 $A E$ 是 $\\angle B A C$ 的角平分线. 又 $A D \\perp O_{1} O_{2}$, 于是 $U, V$ 关于直线 $A D$ 对称, 因此 $\\triangle A U V$ 是等腰三角形.'
 '如图, 设 $\\triangle A B C$ 的外心为 $O$, 连结 $O O_{1}, O O_{2}$, 过点 $O, O_{1}, O_{2}$ 分别作直线 $B C$ 的垂线, 垂足分别为 $D, D_{1}, D_{2}$. 作 $O_{1} K \\perp O D$ 于点 $K$.\n\n<img_4154>\n\n我们证明 $O O_{1}=O O_{2}$. 在直角三角形 $O K O_{1}$ 中,\n\n$$\nO O_{1}=\\frac{O_{1} K}{\\sin \\angle O_{1} O K}\n$$\n\n由外心的性质, $O O_{1} \\perp A C$, 又 $O D \\perp B C$, 故 $\\angle O_{1} O K=\\angle A C B$. 而 $D, D_{1}$ 分别是 $B C, C X$ 的中点, 所以\n\n$$\nD D_{1}=C D_{1}-C D=\\frac{1}{2} C X-\\frac{1}{2} B C=\\frac{1}{2} B X\n$$\n\n因此\n\n$$\nO O_{1}=\\frac{O_{1} K}{\\sin \\angle O_{1} O K}=\\frac{D D_{1}}{\\sin \\angle A C B}=\\frac{\\frac{1}{2} B X}{\\frac{A B}{2 R}}=R \\cdot \\frac{B X}{A B}\n$$\n\n\n\n这里 $R$ 是 $\\triangle A B C$ 的外接圆半径. 同理 $O O_{2}=R \\cdot \\frac{C Y}{A C}$. 由已知条件可得 $\\frac{B X}{A B}=\\frac{C Y}{A C}$,故 $O O_{1}=O O_{2}$. 由于 $O O_{1}=O O_{2}$, 故 $\\angle O O_{1} O_{2}=\\angle O O_{2} O_{1}$, 从而 $\\angle A U V=\\angle A V U$.这样 $A U=A V$, 即 $\\triangle A U V$ 是等腰三角形.']"			['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIATICIQMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiig9KACiuX8deM7PwL4cfVbmLz5C4jgtg+0yue2cHAxkk4PSuht5VngilU5WRQyn1BGRQBPRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABSEgKSTgAdaU9K4/xldXt3HH4f0q1e6uLpfMvFSUIUtc4YbjwC/KjvjeR0oA8/wDitanxF8PtV8USg/Z4ZIo9MU9oPNVWl+sjEEf7Kp6mvVPB9x9s8F6FdE5MunwOfqY1J/WuI+I02uXfw11mzl8MC1tltQS63sbCNYyGyFA5A29K3/hNc/avhd4fkznFt5Wf9xmX+lAHaUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABQehoooAy9avr+xs/M07R59TnJ4iimjix7szsMD6A/Suf8CJr2NTufEmitY6ldz+c0vnxSI6fdSNdrFgEUdCOpJ7mu0ooA5bx1/adz4a1DS9M0O41KW/tJrfek0UaRFk2gtvYHvn5QenasT4UWPiDw54YtPDutaHPbfZzKy3QuIXjIZywGFctnLHtXolFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFIxARiSAAOSegoAU9DXP8Airxjovg/TTe6veLECD5cK4Msp9FXv9eg71yOtfEufULq50zwPBDqM9upa71S4fbY2S92Z+jYAJ447jOCK4jwH4El8d+KpPFOvXdxqelwSYinuUKfbnU/wp0WFT0Xv0I6gAHrngjxzpXjrS5L3TfNjeGTy5oJwBIhIyOAT8p5wfY+hrqa+b9Ca88CeJtd8UW29tNtNcn0/VbRB922ZgYpVUD+Ek/mB0Jr6Jtp4rq2iuIHWSKVQ6OhyGUjIIPcEYoAnooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKD0pGICEkgADqegrzXVPiJe61qEuh/D+0XVb5eJ9Rc4tLTPct0c4zwPw3cigDq/FPjDRfCFh9q1a7EbP8A6qBPmlmPoijk/XpXEroviv4mMJfELTeH/DLEMulQti4uh285uwPp+mQGrd8L/Dm10jUW1vWrqTW/EMpy99cjiP2jToo7Dv6YHAzPGWv6h4l1z/hBPC02y4cbtW1BTlbOE8FB/tn07dPUqAYzWdr451AeDfC8K2PgnSnA1K5tuBdyA58lD1PQktk56n+Hd65Z2lvYWsNraQxw28KCOOONdoVR0AHoKp6BoWn+G9HttK0yHyrWAYAxyx7sx7sT1NatAHmXh2zgl+J3xF0W7i820vY7SUxP0dWiIf8AMtj8Kb4AvLjwn4huvh5qszOsANzo88h5ntic7Pqpzx7NjgCrP/Hl+0GvaO/8P/8AfTrN/wDEj9a0PiN4Xude0aG/0hjFr+lSfarCVepYctH9Gx06ZA7UAdtRXO+C/FNv4w8NWurQARyP8lxDnmGYfeQ/0zzgg10VABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUU12VI2Z2CqASSegFADj0rnvFXjLRvCFkLjVbra78Q28fzSzH0VRyfTPQVyuqfEO/wBdv5dD+H1mmo3qHbPqcvFpa++7o5+nHpurT8LfDqz0XUf7a1a6l1rxBIcvf3YzsPpGvIQY49ewIHFAGEmieK/iU63HiUy6F4bJDJpEDkT3I/6bN1A9v0BG4+jaVpGn6Hp8dhplnFa20YwscSgD6+59zV+uS8deMofCGlxGKE3WrXr+Tp9kmS08h4HA52gkZx6gd6AM/wAe+LrzTp7Xw14cRbjxLqYxAp+7bR95X9hg49wT2wdfwX4PsvB2jCzhZri7lbzby8l5e4lPJYn09B/Mkk5/gLwbN4eS51bWZvtviXU28y+ujztHaNPRRjHHt2Ax21ABR2oooA858Yf6J8W/AN7khZTeWznPXMY2j8yfyr0Y9K84+LJ+zQ+E9TGR9j8QWpd84whzu/A4Ar0XHv8AiKAPLNV3fDX4hprifL4b8QSiHUAPu211ztl9g3JP/AvYV6mCDg8HPPHNZuu6JZeI9CvNI1CPdbXUZRgOqnqGB/vA4I9xXHfDPW723a88E644bV9EwkchwPtNrx5ci+uBgH6rk5zQB6LRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABQelNd1jjZ3YKqgksxwAPevNdR+IWoeIb6XRPh7aLf3KHZPq0v/Hpa+4P8bYzjHHf5hQB1XirxnonhC0WfVLrbLJ/qbaIb5pj/ALK9T6Z6Vxi6F4s+JMon8TGXQfDmQU0iByJ7kdf3zdh04/QEZO94V+HVjoV+dZ1O5l1nxBLzLf3XJU+kanOwfTntkDiu3oAo6VpNhomnxWGmWkVraxfdiiUAD39yfWr1HaqOp6nZ6Nps9/qFwtva26F5JHPAH9T6CgCj4p8Taf4R0KbVdRfEUZ2oin55X/hRQerHn8AT2rlfA3hrUb/Vn8c+K4h/bV2m2ztWHGnwEcIAejEE59MnPJYVQ8NabefEXxHD40123eHR7VidE0+TuM/69x6kgEfpwAW9UAx06UALRRRQAUUUUAcB8aLd5/hXqzxZ82AwzoQeRtlXJ/Imu3srlLyyt7mM5jmjWRT6gjNY3jm0N74B8QWyjLvp8+0erbCR+uKh+Hl0L34deHZg27/QIUYnn5lUKf1U0AdPXnfxJ0W9hNn4z0FCdY0T53jH/LzbdXjPrxkj8cc4r0Sg9DQBleH9bsvEWh2er2D77e6jDrzkqehU47g5B+hrVryrSz/wrX4iHRmynhnxFIZbE/w210fvR/RuMfVfQmvVM0ALRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRTZHSOJ5JGVUVSWZjgADuaAHHpXO+KvGWi+ELNZtTucSyf6m1iG+ac+ir1PpnoK5XUfiDqfiPUJdE+Hlql9PH8lxrE+Ra2/OODj52+nHcBh01fCvw7sNDv21jUrmXWtflOZNRuxkg/9M15CDHHH0BxxQBgpoHir4kutz4pMmieHSQ0ejQORNcDt57dQPb9ARk+kaXpdjo2nxWOnWsVtaxDCRxIFA9/r71dooAKKKKAGyOkUTySMqIoLMzHAAHUk15Mgm+LviMNlk8D6XPkAjB1Ocf8AtMH8/qflm8Q6jd/EbxFN4M0Sd4tFtDjW9QQ4Lf8ATBCOMkgg/iOgIb0jTdOtNJ0+3sLC3WC1t0CRxoMBV/r/AJ70AWY41iRURQqKAAo7Y4qSiigAooooAKKKKAGTRLPBJE4yjqVYexGK4D4KyP8A8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proof	Plane Geometry	Math	Chinese
9	"如图, $\triangle A B C$ 内接于圆 $O, P$ 为弧 $B C$ 上一点, 点 $K$ 在线段 $A P$ 上, 使得 $B K$ 平分 $\angle A B C$. 过 $K 、 P 、 C$ 三点的圆 $\Omega$ 与边 $A C$ 交于点 $D$, 连接 $B D$ 交圆 $\Omega$ 于点 $E$, 连接 $P E$ 并延长与边 $A B$ 交于点 $F$, 证明: $\angle A B C=2 \angle F C B$.

<image_1>"	"['设 $C F$ 与圆 $\\Omega$ 交于点 $L$ (异于 $C$ ). 连接 $P B 、 P C 、 B L 、 K L$, 如下图.\n\n<img_4215>\n\n注意此时 $C 、 D 、 L 、 K 、 E 、 P$ 六点均在圆 $\\Omega$ 上, 结合 $A 、 B 、 P 、 C$ 四点共圆, 可知\n\n$$\n\\angle F E B=\\angle D E P=180^{\\circ}-\\angle D C P=\\angle A B P=\\angle F B P\n$$\n\n因此 $\\triangle F B E \\sim \\triangle F P B$, 因此\n\n$$\nF B^{2}=F E \\cdot F P\n$$\n\n又由圆幂定理可知, $F E \\cdot F P=F L \\cdot F C$, 所以\n\n$$\nF B^{2}=F L \\cdot F C,\n$$\n\n\n\n因此 $\\triangle F B L \\sim \\triangle F C B$. 因此\n\n$$\n\\angle F L B=\\angle F B C=\\angle A P C=\\angle K P C=\\angle K L C,\n$$\n\n即 $B 、 K 、 L$ 三点共线. 再根据 $\\triangle F B L \\sim \\triangle F C B$ 得\n\n$$\n\\angle F C B=\\angle F B L=\\angle F B K=\\frac{1}{2} \\angle A B C,\n$$\n\n即 $\\angle A B C=2 \\angle F C B$.'
 '设 $C F$ 与圆 $\\Omega$ 交于点 $L$ (异于 $C$ ), 如下图.\n\n<img_4180>\n\n对圆内接广义六边形 $D C P E L K$ 应用帕斯卡定理可知, $D C$ 与 $K P$ 的交点 $A 、 C L$ 与 $P E$ 的交点 $F 、 L K$ 与 $E D$ 的交点 $B^{\\prime}$ 共线, 因此 $B^{\\prime}$ 是 $A F$ 与 $E D$ 的交点, 即 $B^{\\prime}=B$,所以 $B 、 K 、 L$ 共线. 根据 $A 、 B 、 P 、 C$ 四点共圆及 $L 、 K 、 P 、 C$ 四点共圆, 得\n\n$$\n\\angle A B C=\\angle A P C=\\angle K L C=\\angle F C B+\\angle L B C,\n$$\n\n又由 $B K$ 平分 $\\angle A B C$ 知\n\n$$\n\\angle A B C=2 \\angle L B C\n$$\n\n从而 $\\angle A B C=2 \\angle F C B$.']"			['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proof	Plane Geometry	Math	Chinese
10	"平面直角坐标系中 $x O y$ 中, $P$ 是不在 $x$ 轴上的一个动点, 过 $P$ 作抛物线 $y^{2}=4 x$ 的两条切线, 切点设为 $A, B$, 且直线 $P O \perp A B$ 于 $Q, R$ 为直线 $A B$ 与 $x$ 轴的交点.

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求证: $R$ 是定点;"	['设 $P(m, n)$, 则 $A B: n y=2(x+m)$, 根据题意, 直线 $P O$ 与直线 $A B$ 垂直, 于是\n\n$$\n\\frac{n}{m} \\cdot \\frac{2}{n}=-1\n$$\n\n因此 $m=-2$. 进而 $R(2,0)$ 为定点.']			['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proof	Plane Geometry	Math	Chinese
11	"如图, 在锐角三角形 $A B C$ 中, $\angle B A C \neq 60^{\circ}$, 过点 $B, C$ 分别作三角形 $A B C$ 的外接圆的切线 $B D, C E$, 且满足 $B D=C E=B C$. 直线 $D E$ 与 $A B, A C$ 的延长线分别交于点 $F, G$. 设 $C F$ 与 $B D$ 交于点 $M, C E$ 与 $B G$ 交于点 $N$.

<image_1>

证明: $A M=A N$."	['如图, 设两条切线 $B D, C E$ 交于点 $K$, 则 $B K=C K$. 结合 $B D=C E$ 可知 $D E \\| B C$.作 $\\angle B A C$ 的平分线 $A L$ 交 $B C$ 于点 $L$, 连结 $L M, L N$.\n\n<img_4245>\n\n由 $D E \\| B C$ 知,\n\n$$\n\\angle A B C=\\angle D F B, \\angle F D B=\\angle D B C=\\angle B A C,\n$$\n\n故 $\\triangle A B C$ 与 $\\triangle D F B$ 相似.\n\n由此并结合 $D E \\| B C, B D=C B$ 及内角平分线定理可得\n\n$$\n\\frac{M C}{M F}=\\frac{B C}{F D}=\\frac{B D}{F D}=\\frac{A C}{A B}=\\frac{L C}{L B},\n$$\n\n因此 $L M \\| B F$.\n\n同理, $L N \\| C G$.\n\n\n\n由此推出\n\n$$\n\\begin{aligned}\n\\angle A L M & =\\angle A L B+\\angle B L M \\\\\n& =\\angle A L B+\\angle A B L \\\\\n& =180^{\\circ}-\\angle B A L \\\\\n& =180^{\\circ}-\\angle C A L \\\\\n& =\\angle A L C+\\angle A C L \\\\\n& =\\angle A L C+\\angle C L N \\\\\n& =\\angle A L N,\n\\end{aligned}\n$$\n\n再结合 $B C \\| F G$ 及内角平分线定理可得\n\n$$\n\\begin{aligned}\n\\frac{L M}{L N} & =\\frac{L M}{B F} \\cdot \\frac{B F}{C G} \\cdot \\frac{C G}{L N} \\\\\n& =\\frac{C L}{B C} \\cdot \\frac{A B}{A C} \\cdot \\frac{B C}{B L} \\\\\n& =\\frac{L C}{L B} \\cdot \\frac{A B}{A C}=1,\n\\end{aligned}\n$$\n\n即 $L M=L N$.\n\n故由\n\n$$\nA L=A L, \\angle A L M=\\angle A L N, L M=L N,\n$$\n\n得到 $\\triangle A L M$ 与 $\\triangle A L N$ 全等, 因而 $A M=A N$, 证毕.']			['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']	Multimodal	Competition	False				Theorem proof	Plane Geometry	Math	Chinese
12	"如图, $A B$ 是圆 $\omega$ 的一条弦, $P$ 为弧 $A B$ 内一点, $E, F$ 为线段 $A B$上两点, 满足 $A E=E F=F B$. 连结 $P E, P F$ 并延长, 与圆 $\omega$ 分别相交于点 $C, D$. 求证:

$$
E F \cdot C D=A C \cdot B D
$$

<image_1>"	['连结 $A D, B C, C F, D E$, 如下图. \n\n<img_4113>\n\n\n由于 $A E=E F=F B$, 从而\n\n$$\n\\frac{B C \\cdot \\sin \\angle B C E}{A C \\cdot \\sin \\angle A C E}=\\frac{\\text { 点 } B \\text { 到直线 } C P \\text { 的距离 }}{\\text { 点 } A \\text { 到直线 } C P \\text { 的距离 }}=\\frac{B E}{A E}=2 . (1)\n$$\n\n同样\n\n$$\n\\frac{A D \\cdot \\sin \\angle A D F}{B D \\cdot \\sin \\angle B D F}=\\frac{\\text { 点 } A \\text { 到直线 } P D \\text { 的距离 }}{\\text { 点 } B \\text { 到直线 } P D \\text { 的距离 }}=\\frac{A F}{B F}=2 . (2)\n$$\n\n另一方面, 由于\n\n$$\n\\begin{aligned}\n& \\angle B C E=\\angle B C P=\\angle B D P=\\angle B D F, \\\\\n& \\angle A C E=\\angle A C P=\\angle A D P=\\angle A D F,\n\\end{aligned}\n$$\n\n\n\n故将 (1) 、(2) 两式相乘可得 $\\frac{B C \\cdot A D}{A C \\cdot B D}=4$, 即\n\n$$\nB C \\cdot A D=4 A C \\cdot B D . (3)\n$$\n\n由托勒密定理\n\n$$\nA D \\cdot B C=A C \\cdot B D+A B \\cdot C D . (4)\n$$\n\n故由 (3)(4) 得\n\n$$\nA B \\cdot C D=3 A C \\cdot B D\n$$\n\n即\n\n$$\nE F \\cdot C D=A C \\cdot B D\n$$']			['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']	Multimodal	Competition	False				Theorem proof	Plane Geometry	Math	Chinese
13	"如图在平面直角坐标系 $x O y$ 中, 菱形 $A B C D$ 的边长为 4 , 且 $|O B|=|O D|=6$.

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求证: $|O A| \cdot|O C|$ 为定值;"	['因为\n\n$$\n|O B|=|O D|,|A B|=|A D|=|C B|=|C D| \\text {, }\n$$\n\n所以 $O, A, C$ 三点共线. \n\n\n连接 $B D$, 则 $B D$ 垂直平分线段 $A C$, 设垂足为 $K$, 如下图.\n\n<img_4132>\n\n于是有\n\n$$\n\\begin{aligned}\n|O A| \\cdot|O C| & =(|O K|-|A K|)(|O K|+|A K|) \\\\\n& =|O K|^{2}-|A K|^{2} \\\\\n& =\\left(|O B|^{2}-|B K|^{2}\\right)-\\left(|A B|^{2}-|B K|^{2}\\right) \\\\\n& =|O B|^{2}-|A B|^{2} \\\\\n& =20 .\n\\end{aligned}\n$$']			['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAa8BmAMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAopMijcPWgBaKM84ooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigApGxjmlpD0oA838ceKNU0rwtqviGzvns0tLgWtjbiJGF04kCMX3KWxkOAFK8LuycgDpfEniGbQPAl9rksapdQWXmCM/dEpGFU+24gVy3xEibVvH3gPQJR/oU17NeSqfuu0KhgPyLD8aZ8ZZzf6XofhWKQefrepxQuoPPlKwLHHsSlAHa+ELvUNQ8I6Te6ps+23FrHNLsXaMsA3TtwRn3zW3TI0WJFRFCqoCqB0AHalkkSKNndgqqMknsKAHUU2ORJUV0YMrAMCO4NOoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACmv09u9OpGGR0z+NAHDSXdt4w8X6hol3oVzHBobpJDq/mtGROQp2oRgj5X9SCMZGCKwdF0S28SfFdfEFkss2kaLC0MV7LK0hurk5DbWY8ogJHGFDZx1r0O/8AD2lalKZrvToJ5GAV2deXUdA394cng8c+9aEEEdtEkUMSRRIoVI41CqoHQADpQBIOPauG+LviJdA+G+pukmLi8T7HAB1JkyDj0ITeQfau5bOOK8r+MvgW68T6PNqra39ms9HsprlbMW27zXVSxJbeMZCgDg459aANr4QeIV8Q/DjTHZw1xZp9jmHcGMAKT9V2n8a7uvLPgz4HuvC+jxaqutfaLPWLKG5ayNtt8qRlDAht5zgMVPyjPHpXqdABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABXP8Ajv8A5J74l/7BV1/6Kaugrn/Hf/JPfEv/AGCrr/0U1AB4E/5J74a/7BVr/wCiVroK5/wJ/wAk98Nf9gq1/wDRK10FABRRRQAUUUUAFFFFABRRRQAUUUUAFBOOtFQ3TTpayNbRLLOFJSN32Bj2BbBx9cGgCXNLkV5xYfETXtQ8ZX/heDwpam/sYxLO/wDav7sKQpHzeTkn5xxj19K1dU8T+JdD06bUb/wnFJZ26mSb7FqQlkRB1O1o1zgc9aAOyorJ8O+INO8UaNBq+ly+bazA4JGGUg4KsOxB7VrUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABXP+O/+Se+Jf8AsFXX/opq6Cuf8d/8k98S/wDYKuv/AEU1AB4E/wCSe+Gv+wVa/wDola6Cuf8AAn/JPfDX/YKtf/RK10FABRRRQAUUUUAFFFFABRRRQAUUUUAFI7BFLMcKOSfalrk/iZrH9ifDjXLxWw5tjAh7hpCIwfw3Z/CgDifg1eW19eeLPFd3cxRSapqJWNZXAKxrlhjPb5wP+A11PjfxdB/Y13o2gJ/a+t3kTQRW1p+8EW8Fd8jD5UUA5+YjP0yQz4a+FrDTvhzocVzYW0k72wnkaSFSxMh34JI7bsfhXOfGWO18K6BZeINEZNM1eO9RI5bQCMyqQxZXA4YcZ59MUAdf8M/CEvgrwbb6XcyiS6Zmnn2nKq7Yyo9gAB74zXYVS0m4nu9Jsrm5h8iea3SSSLGNjFQSvPoTirtABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAVz/jv/AJJ74l/7BV1/6Kaugrn/AB3/AMk98S/9gq6/9FNQAeBP+Se+Gv8AsFWv/ola6Cuf8Cf8k98Nf9gq1/8ARK10FABRRRQAUUUUAFFFFABRRRQAUUUUAIwJGBXmfxG8J+MfHGmjSLePRbXT0uhMXe9lMkyrkKCBDheuSBu5A54r02igDl1l8ZpCscWi+H0VVCqDqkxAA6cfZ/61ir8Pb7X/ABDb6540v4b5rRs2em2qlbaDpkkty54B5x0HbgehE4GTSAg9DQAijFOoooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACuf8d/8AJPfEv/YKuv8A0U1dBXP+O/8AknviX/sFXX/opqADwJ/yT3w1/wBgq1/9ErXQVz/gT/knvhr/ALBVr/6JWugoAKKKKACiiigAooooAKKKKACiiigAo6UVHPNFBC8s0ixxRgu7scBQOSSaAOY8cePdK8D2ljNqCvKby4ESRxfe2fxvjuFBHHckDvXQ2N7bX9lDe2k8c1rOgeKVDkMD0Oa8j1qKTWvCWteP7uIgs1u+lxyKf3NnFOj7sdd0m0sepxt5GK6WaOT4b6k95ArHwjdybriFRn+zZWP+sXniJiTuGMKcEYBNAHoIOaKjikSSNXR1ZGAKsGyCPY96kBzQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFc/wCO/wDknviX/sFXX/opq6Cuf8d/8k98S/8AYKuv/RTUAHgT/knvhr/sFWv/AKJWugrn/An/ACT3w1/2CrX/ANErXQUAFFFFABRRRQAUUUUAFFFFABRRRQAjHAzXFeL5n8QavaeCrV2Edyn2nVZE48u0Bxsz/ekb5fXburpNd1i00HQ7vVr1iLe2jMjberdgq+5JAHuax/BGj3VnY3OrasoGt6xILq8Uf8shjEcQ9kXA+u40AS+N7QP8O/EFrFGqqumTiNFAwMRtgAdugxWnpssWq+H7SZ40eK6tUcqyhgVdM4PYjBq1fW4u9PubZuVliaM59wRXP/Dm4W6+HHh2RSCBp8MfBz91Qv8ASgDHtJpfh5qsWlXUjSeFrx9lhcyMSbCQ9IHP/PMn7rH7v3ScYrvgeSO9V9R0601XT57C/t0uLW4QpLE4yGB/z17VyGh6jd+EtXh8La5cNNZzfLpGpSn/AFo/595D/wA9QOh/iA9RyAdzRSAg9KWgAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigArn/AB3/AMk98S/9gq6/9FNXQVz/AI7/AOSe+Jf+wVdf+imoAPAn/JPfDX/YKtf/AEStdBXP+BP+Se+Gv+wVa/8Aola6CgAooooAKKKKACiiigAooooAKQ0EgDk4rA8Y6+/h/QWltYvP1K5kW1sLcf8ALWd+FH0HLH2U0AYt6P8AhMPHUenL82j6BItxdEdJrsjMcfoQg+c+5UV3Crg/yrH8K6BH4b0GDTxIZp8mW5nPWeZjudz9WJ/DA7VtUAI3SuS+Gq+R4OWz3Fvsd7eW/JzgLcyAD8sV1prj/AQaGbxVaMT+5164ZQTnCyKkv83NAHYGs3XdDsvEOkTabqEe6GUdQcMjDo6nswPIPrWnSHOOOvvQBx3hjXL2x1M+FfEcu7Uo0L2V4eF1CEfxD0kXgMv4jINdlWH4l8NWviTS/s00jW9zE4ltLuP/AFlvMBhXU+o9O9UPCniO4u7i50PXUW38QWK5nQDCXMWSFni9UOOR/CcggdwDq6KQEHpS0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAVz/AI7/AOSe+Jf+wVdf+imroK5/x3/yT3xL/wBgq6/9FNQAeBP+Se+Gv+wVa/8Aola6Cuf8Cf8AJPfDX/YKtf8A0StdBQAUUUUAFFFFABRRRQAUUUh96AEY9PrXDaAP+Ev8YXHiZ8PpWmb7LSeQRI+cTTj8RsU56A+tXPHGo3UsVp4Z0uXZqeskxeavW3tx/rZfqFOB0+Zh6Guj0zT7bSdOttOsoxHa20SxRp6KoAH1+tAFsDBpaKKAA9K4/wANF4vH/jS1PCtLaXS8f34Qh/WI117ZxxXI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']	Multimodal	Competition	False				Theorem proof	Plane Geometry	Math	Chinese
14	"如图, 在锐角 $\triangle A B C$ 中, $A B>A C, M, N$ 是 $B C$ 边上不同的两点, 使得 $\angle B A M=$ $\angle C A N$. 设 $\triangle A B C$ 和 $\triangle A M N$ 的外心分别为 $O_{1}, O_{2}$, 求证: $O_{1}, O_{2}, A$ 三点共线.

<image_1>"	['连接 $A O_{1}, A O_{2}$, 过 $A$ 作 $A O_{1}$ 的垂线 $A P$ 交 $B C$ 的延长线于点 $P$, \n如下图.\n\n<img_4220>\n\n\n则有 $A P$ 是圆 $O_{1}$ 的切线, 所以\n\n$$\n\\angle B=\\angle P A C\n$$\n\n又因为 $\\angle B A M=\\angle C A N$, 所以\n\n$$\n\\angle A M P=\\angle B+\\angle B A M=\\angle P A C+\\angle C A N=\\angle P A N\n$$\n\n因而 $A P$ 是 $\\triangle A M N$ 外接圆 $O_{2}$ 的切线, 故 $A P \\perp A O_{2}$.\n\n因为\n\n$$\nA P \\perp A O_{1}, A P \\perp A O_{2},\n$$\n\n所以 $O_{1}, O_{2}, A$ 三点共线.']			['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proof	Plane Geometry	Math	Chinese
15	"作斜率为 $\frac{1}{3}$ 的直线 $l$ 与椭圆 $C: \frac{x^{2}}{36}+\frac{y^{2}}{4}=1$ 交于 $A, B$ 两点 (如图所示), 且 $P(3 \sqrt{2}, \sqrt{2}$ )在直线 $l$ 的左上方.


<image_1>
证明: $\triangle P A B$ 的内切圆的圆心在一条定直线上;"	['设直线 $l: y=\\frac{1}{3} x+m, A\\left(x_{1}, y_{1}\\right), B\\left(x_{2}, y_{2}\\right)$.\n\n将 $y=\\frac{1}{3} x+m$ 代入 $\\frac{x^{2}}{36}+\\frac{y^{2}}{4}=1$ 中, 化简得\n\n$$\n2 x^{2}+6 m x+9 m^{2}-36=0 .\n$$\n\n于是有\n\n$$\nx_{1}+x_{2}=-3 m, x_{1} x_{2}=\\frac{9 m^{2}-36}{2} .\n$$\n\n因为\n\n$$\nk_{P A}=\\frac{y_{1}-\\sqrt{2}}{x_{1}-3 \\sqrt{2}}, k_{P B}=\\frac{y_{2}-\\sqrt{2}}{x_{2}-3 \\sqrt{2}}\n$$\n\n所以\n\n$$\n\\begin{aligned}\nk_{P A}+k_{P B} & =\\frac{y_{1}-\\sqrt{2}}{x_{1}-3 \\sqrt{2}}+\\frac{y_{2}-\\sqrt{2}}{x_{2}-3 \\sqrt{2}} \\\\\n& =\\frac{\\left(y_{1}-\\sqrt{2}\\right)\\left(x_{2}-3 \\sqrt{2}\\right)+\\left(y_{2}-\\sqrt{2}\\right)\\left(x_{1}-3 \\sqrt{2}\\right)}{\\left(x_{1}-3 \\sqrt{2}\\right)\\left(x_{2}-3 \\sqrt{2}\\right)},\n\\end{aligned}\n$$\n\n上式中\n\n$$\n\\begin{aligned}\n\\text { 分子 } & =\\left(\\frac{1}{3} x_{1}+m-\\sqrt{2}\\right)\\left(x_{2}-3 \\sqrt{2}\\right)+\\left(\\frac{1}{3} x_{2}+m-\\sqrt{2}\\right)\\left(x_{1}-3 \\sqrt{2}\\right) \\\\\n& =\\frac{2}{3} x_{1} x_{2}+(m-2 \\sqrt{2})\\left(x_{1}+x_{2}\\right)-6 \\sqrt{2}(m-\\sqrt{2}) \\\\\n& =\\frac{2}{3} \\cdot \\frac{9 m^{2}-36}{2}+(m-2 \\sqrt{2})(-3 m)-6 \\sqrt{2}(m-\\sqrt{2}) \\\\\n& =3 m^{2}-12-3 m^{2}+6 \\sqrt{2} m-6 \\sqrt{2} m+12=0,\n\\end{aligned}\n$$\n\n因此\n\n$$\nk_{P A}+k_{P B}=0\n$$\n\n\n\n又 $P$ 在直线 $l$ 的左上方, 因此, $\\angle A P B$ 的角平分线是平行于 $y$ 轴的直线, 所以 $\\triangle P A B$的内切圆的圆心在直线 $x=3 \\sqrt{2}$ 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proof	Plane Geometry	Math	Chinese
16	"如图, $P, Q$ 分别是圆内接四边形 $A B C D$ 的对角线 $A C, B D$ 的中点. 若 $\angle B P A=$ $\angle D P A$, 证明: $\angle A Q B=\angle C Q B$.

<image_1>"	['延长线段 $D P$ 与圆交于另一点 $E$, 延长线段 $A Q$ 与圆交于另一点 $F$, 如下图. \n\n<img_4167>\n\n则\n\n$$\n\\angle C P E=\\angle D P A=\\angle B P A .\n$$\n\n因为 $P$ 是线段 $A C$ 的中点, 故 $\\widehat{A B}=\\widehat{C E}$, 从而\n\n$$\n\\angle C D P=\\angle B D A \\text {. }\n$$\n\n又 $\\angle A B D=\\angle P C D$, 所以\n\n$$\n\\triangle A B D \\backsim \\triangle P C D\n$$\n\n于是\n\n$$\n\\frac{A B}{B D}=\\frac{P C}{C D},\n$$\n\n即\n\n$$\nA B \\cdot C D=P C \\cdot B D,\n$$\n\n从而有\n\n$$\n\\begin{aligned}\nA B \\cdot C D & =\\frac{1}{2} A C \\cdot B D \\\\\n& =A C \\cdot\\left(\\frac{1}{2} B D\\right) \\\\\n& =A C \\cdot B Q,\n\\end{aligned}\n$$\n\n又 $\\angle A B Q=\\angle A C D$, 所以 $\\triangle A B Q \\backsim \\triangle A C D$, 故\n\n$$\n\\angle Q A B=\\angle D A C .\n$$\n\n同时, 有 $\\angle C A B=\\angle D A F$, 故\n\n$$\n\\widehat{B C}=\\widehat{D F}.\n$$\n\n又因为 $Q$ 为 $B D$ 的中点, 所以\n\n$$\n\\angle C Q B=\\angle D Q F,\n$$\n\n而 $\\angle A Q B=\\angle D Q F$, 所以\n\n$$\n\\angle A Q B=\\angle C Q B.\n$$']			['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']	Multimodal	Competition	False				Theorem proof	Plane Geometry	Math	Chinese
17	"<image_1>

如图, 圆 $O 1$ 和圆 $O 2$ 与 $\triangle A B C$ 的三边所在的三条直线都相切, $E 、 F 、 G 、 H$ 为切点, 并且 $E G 、 F H$ 的延长线交于 $P$ 点。求证直线 $P A$ 与 $B C$ 垂直。"	['证明 设 $\\triangle A B C$ 的三边分别为 $a、b、c$, 三个角分别为 $A、 B 、 C$ 则 $C E=B F=C G=B H=\\frac{1}{2}(a+b+c)$.\n\n$\\therefore B E=\\frac{1}{2}(a+b+c)-a=\\frac{1}{2}(b+c-a)$.\n\n$\\therefore E F=\\frac{1}{2}(a+b+c)+\\frac{1}{2}(b+c-a)=b+c$.\n\n连 $C O_{2}$, 则 $C O_{2}$ 平分 $\\angle E C G, CO_{2} \\perp E G \\Rightarrow \\angle F E P=90^{\\circ}-\\frac{1}{2}$ $\\angle C$.\n\n同理 $\\angle E F P-90^{\\circ}-\\frac{1}{2} \\angle B, \\angle E P F=\\frac{1}{2}(B+C)$.\n\n<img_4224>\n\n$$\n\\because \\frac{E P}{\\sin \\left(90^{\\circ}-\\frac{1}{2} B\\right)}=\\frac{E F}{\\sin \\frac{B^{+} C}{2}} \\quad, \\quad \\therefore\n$$\n\n$E P=(b+c) \\frac{\\cos \\frac{B}{2}}{\\sin \\frac{B+C}{2}}$.\n\n设 $P 、 A$ 在 $E F$ 上的射影分别为 $M 、 N$, 则 $E M=E P \\cos \\angle F E P=(b+c) \\frac{\\cos \\frac{B}{2} \\sin \\frac{C}{2}}{\\sin \\frac{B+C}{2}}$.\n\n又 $B N=c \\cos B$, 故只须证 $c \\cos B+\\frac{1}{2}(b+c-a)=(b+c) \\frac{\\cos \\frac{B}{2} \\sin \\frac{C}{2}}{\\sin \\frac{B+C}{2}}$,\n\n即 $\\sin C \\cos B+\\frac{1}{2}(\\sin B+\\sin C-\\sin (B+C))=\\frac{\\cos \\frac{B}{2} \\sin \\frac{C}{2}}{\\sin \\frac{B+C}{2}}(\\sin B+\\sin C)$ 就是 $2 \\cos \\frac{B-C}{2} \\cos \\frac{B}{2} \\sin \\frac{C}{2}=\\sin C \\cos B-\\frac{1}{2} \\sin B \\cos C-\\frac{1}{2} \\cos B \\sin C+\\sin \\frac{B+C}{2} \\cos \\frac{B-C}{2}$\n\n右边 $=\\frac{1}{2} \\sin (C-B)+\\sin \\frac{B+C}{2} \\cos \\frac{B-C}{2}=\\cos \\frac{B-C}{2}\\left(\\sin \\frac{B+C}{2}-\\sin \\frac{B-C}{2}\\right)$ $=2 \\cos \\frac{B-C}{2} \\cos \\frac{B}{2} \\sin \\frac{C}{2}$ 。故证。']			['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']	Multimodal	Competition	False				Theorem proof	Geometry	Math	Chinese
18	"如图, 在四边形 $A B C D$ 中，对角线 $A C$ 平分 $\angle B A D$ 。在 $C D$ 上取一点 $E, B E$ 与 $A C$ 相交于 $F$, 延长 $D F$ 交 $B C$ 于 $G_{\circ}$ 求证: $\angle G A C=\angle E A C$.

<image_1>"	['连结 $\\mathrm{BD}$ 交 $\\mathrm{AC}$ 于 $\\mathrm{H}$. 对 $\\triangle \\mathrm{BCD}$ 用塞瓦定理, 可得 $\\frac{C G}{G B} \\cdot \\frac{B H}{H D} \\cdot \\frac{D E}{B C}=1$因为 $A H$ 是 $\\angle \\mathrm{B} A \\mathrm{D}$ 的平分线, 由角平分线定理, 可得 $\\frac{B H}{H D}=\\frac{A B}{A D}$.\n\n敬 $\\frac{C G}{G B} \\cdot \\frac{A B}{A D} \\cdot \\frac{D E}{B C}=1$.\n\n过点 $\\mathrm{C}$ 作 $A B$ 的平行线 $A G$ 的延长线于 $I$, 过点 $C$ 作 $A D$ 的平行线交 $A \\mathrm{E}$ 的延长线于 $\\mathrm{J}$.\n\n则 $\\frac{C G}{G B}=\\frac{C I}{A B}, \\frac{D E}{E C}=\\frac{A D}{C J}$. 所以, $\\frac{C I}{A B} \\cdot \\frac{A B}{A D} \\cdot \\frac{A D}{C J}=1$\n\n从而, $\\mathrm{CI}=\\mathrm{CJ}$.\n\n又因为 $\\mathrm{CI} / / \\mathrm{AB}, \\mathrm{CJ} / / \\mathrm{AD}$,\n\n故 $\\angle \\mathbf{A C I}=\\boldsymbol{\\pi}-\\angle \\mathbf{A B C}=\\boldsymbol{\\pi}-\\angle \\mathrm{DAC}=\\angle \\mathrm{ACJ}$.\n\n因此, $\\triangle \\mathrm{ACI} \\cong \\triangle \\mathrm{ACJ}$.\n\n从而, $\\angle \\mathrm{IAC}=\\angle \\mathrm{JAC}$, 即 $\\angle \\mathrm{GAC}=\\angle 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proof	Geometry	Math	Chinese
19	"如图, $\triangle A B C$ 和 $\triangle A D E$ 是两个不全等的等腰直角三角形, 现固定 $\triangle A B C$, 而将 $\triangle A D E$绕 $A$ 点在平面上旋转, 试证: 不论 $\triangle A D E$ 旋转到什么位置, 线段 $E C$ 上必存在点 $M$, 使 $\triangle B M D$为等腰直角三角形.

<image_1>"	['以 $A$ 为原点, $A C$ 为 $x$ 轴正方向建立复平面. 设 $C$表示复数 $c$, 点 $E$ 表示复数 $e(c 、 e \\in R)$. 则点 $B$ 表示复数 $b=\\frac{1}{2} c+\\frac{1}{2} c i$,点 $D$ 表示复数 $d=\\frac{1}{2} e-\\frac{1}{2} e i$.\n\n把 $\\triangle A D E$ 绕点 $A$ 旋转角 $\\theta$ 得到 $\\triangle A D E$,\n\n则点 $E^{\\prime}$ 表示复数 $e^{\\prime}=e(\\cos \\theta+i \\sin \\theta)$. 点 $D^{\\prime}$ 表示复数 $d^{\\prime}=d(\\cos \\theta+i \\sin \\theta)$\n\n<img_4086>\n\n表示 $E^{\\prime} C$ 中点 $M$ 的复数 $m=\\frac{1}{2}\\left(c+e^{\\prime}\\right)$.\n\n$\\therefore$ 表示向量 $\\overrightarrow{\\mathrm{MB}}$ 的复数: $z_{1}=b-\\frac{1}{2}\\left(c+e^{\\prime}\\right)=\\frac{1}{2} c+\\frac{1}{2} c i-\\frac{1}{2} \\cdot c-\\frac{1}{2} e(\\cos \\theta+i \\sin \\theta)=-\\frac{1}{2}$ $e \\cos \\theta+\\frac{1}{2}(c-e \\sin \\theta) i$.\n\n表示向量 $\\overrightarrow{\\mathrm{MD}^{\\prime}}$ 的复数: $Z_{2}=d^{\\prime}-m=\\left(\\frac{1}{2} e-\\frac{1}{2} e i\\right)(\\cos \\theta+i \\sin \\theta)-\\frac{1}{2} c-\\frac{1}{2} e(\\cos \\theta+i \\sin \\theta)$\n\n\n\n$$\n=\\frac{1}{2}(e \\sin \\theta-c)-\\frac{1}{2} i e \\cos \\theta .\n$$\n\n显然: $Z_{2}=Z_{1} i$. 于是 $|M B|=|M D^{\\prime}|$, 且 $\\angle B M D^{\\prime}=90^{\\circ}$. 即 $\\triangle B M D^{\\prime}$ 为等腰直角三角形. 故证.']			['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proof	Geometry	Math	Chinese
20	"如图, $M, N$ 分别为锐角三角形 $\triangle A B C(\angle A<\angle B)$ 的外接圆 $\Gamma$ 上弧 $B C 、 A C$ 的中点. 过点 $C$ 作 $P C / / M N$ 交圆 $\Gamma$ 于 $P$ 点, $I$ 为 $\triangle A B C$ 的内心, 连接 $P I$ 并延长交圆 $\Gamma$ 于 $T$.

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求证: $M P \cdot M T=N P \cdot N T$;"	['连 $N I, M I$. 由于 $P C / / M N, P, C, M, N$ 共圆, 故 $P C M N$ 是等腰梯形. 因此 $N P=M C$, $P M=N C$.\n\n<img_4162>\n\n连 $A M ， C I$, 则 $A M$ 与 $C I$ 交于 $I$, 因为\n\n$\\angle M I C=\\angle M A C+\\angle A C I=\\angle M C B+\\angle B C I=\\angle M C I$,\n\n所以 $M C=M I$. 同理\n\n$N C=N I$.\n\n于是\n\n$N P=M I, \\quad P M=N I$.\n\n故四边形 MPNI 为平行四边形。因此 $S_{\\triangle P M T}=S_{\\triangle P N T}$ (同底，等高).\n\n又 $P, N, T, M$ 四点共圆, 故 $\\angle T N P+\\angle P M T=180^{\\circ}$, 由三角形面积公式\n\n$S_{\\triangle P M T}=\\frac{1}{2} P M \\cdot M T \\sin \\angle P M T$\n\n$=S_{\\triangle P N T}=\\frac{1}{2} P N \\cdot N T \\sin \\angle P N T$\n\n$=\\frac{1}{2} P N \\cdot N T \\sin \\angle P M T$\n\n于是 $P M \\cdot M T=P N \\cdot N T$.']			['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAZMBoAMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiisXxP4n0rwno76nq1x5cIOxFUZeVyMhVHc8f16UAbWaMj1rjLHVfGutQLdW2i6dpNtJ80aajM7zsOxZEACHGOCxI70/Ste8Sjxamha3penoslo90l3ZXDspCsilSrLkH5x3/OgDsKKQfSloAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACs7WNc0zQbYXGqXaW0Jz87g449SBxWjXCfGTUf7N+FusMDhp1S3Ueu9wG/wDHd1AF5PiZ4KkYKPEungnoWk2j8zxXUW9xDdQJPbzRzQyKGSSNgysPUEda5fwho1ra/DTR9M1C3ie3Gnxm4jnAKZZcvkH3JrkPgC0x8NawqPI2lpqLrZFyThcAkD25B+pPfNAHrlFFFABRRRQAUVDc3ltZxmS6uIoIwMlpXCgfia5q9+JXgqwBM/ibTTjr5M4lP5Jk0AdXRXnE3xy8AQj5NWmmx/ctJf8A2ZRWa37QfhEyrHb2WtXLN2htkP6FwaAPWaK8q/4XzoP/AEL/AIm/8Ak/+OVG37QXhWOYRXGm67bEjOZrVB+gkzQB6zRmvNbb47eA5wDJqNxb+vm2khx/3yDWzY/FLwNqCgw+JbFc/wDPdjD/AOhgUAdjRVSx1Ow1OIS2F9a3SEZ3W8yyD8wat0AFFGaKAEOOp7V49qf/ABVf7RFppN1l7HQLT7SIW5UyEK2783j/AO+a9hNeTeIrK58GfFxPGxtZ5tF1G2FrqE0EZdrYgKA5Uc7fkTJHoe+MgHq69jxzSGNDIJQimQKVDY5x1xn8BXMyfEXwbHaC4PiXTHQjISO4V5D/AMABLZ9sVN4P1m78QQalqU0M8Fk94UsI54TGxhVFG/BAPzNvPPtQB0Y70tFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAV5H8eWa+0vw54eibEup6oigA8kAbf5yKfyr1mTp1x7+nvXH+I/h5aeJtfstZvNY1aGexcPZxwNAEgb5eQGjJJJUH5ify4oAZP8OoL22+yah4k8RXllja1rJdqkbr/AHW8tFJH411Wl6ZZaNp0Nhp9tHbWsK4SKMYAH+PqTyep5qhe6pYeFdHW41zW8Qpwbm8ZFeQ8nGEVQTjsq9uhryPxR+0Pbxs1r4YsRKSdv229UrGPcRj5j+OOnSgD3R3VULuyhAMkk8AVwmv/ABi8F6Exi/tP+0bjIHk6evnE/wDAshf1rx5NY8OeKHFx4/8AiLeXi53f2fYWk0cK46cmMD8lB967zQfHPwb8MALpLQWzjgTfYZ2k/F2Qt+tAFn/hPfiH4jx/wjHgY2duxwt3qz7cqe4Ulf0LU4eBfiTrg/4n/j/7EhbPk6VFtwPTcAh/PNaY+N/w/H/Mak/8A5v/AIml/wCF4fD7/oNSf+Ac3/xNAFK2+BHhMSmfU59U1aduWa7uiMn/AIAAf1ro7H4YeCdOQJD4Z05wP+fiLzj+b5rJ/wCF4fD7/oNyf+Ac3/xNH/C8Ph9/0G5P/AOb/wCJoA7S10XS7BQlnplnboP4YoEQfoKvAY4GAB2Fee/8Lw+H3/Qbk/8AAOb/AOJo/wCF4fD7/oNyf+Ac3/xNAHoVGP1rz3/heHw+/wCg3J/4Bzf/ABNH/C8Ph9/0G5P/AADm/wDiaAO5uNMsLsbbmyt51PaSJWH61h3nw68G3ykT+GNL+bqY7ZY2P4qAf1rC/wCF4fD7/oNyf+Ac3/xNH/C8Ph9/0G5P/AOb/wCJoAq3/wACPBVzhrSC+06RTlXtbpiQf+B7qqt8N/G+jBn8OfES9bA/d2+pJ5qD2LEsB+C1qf8AC8Ph9/0G5P8AwDm/+Jo/4Xh8Pv8AoNyf+Ac3/wATQBknxT8VPDe4a14Rtdbtk48/S5CHb1O0bif++BWlo/xq8I6hP9lvprnRrtTtaHUItgU/7wyB+OKefjh8P/8AoNyf+Ac3/wATWXrHxL+EniCHy9XngvFwQDNp0pZc/wB1tmVPuCKAPUrW6try2S5tZ4p4JBlZYnDKw9QRwamPX+dfM11d+AtIuHv/AAL491HQ7rO7yHt55YHODhTlM9+rbvpWt4e/aDuLG4+xeI7aHUIlwn2/T1ZWb3KPjP8A479KAPf1t4UkMqwxiQ/xBRn86kAIrD8NeMNB8W2jXOi6jFcBeXj+7JH/ALynkfXpW7QAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFGR60hNcV43+JOkeDwtqVfUNZlwsGnWxy7E9N2M7QcjsSc8A84AOtvr210+1kury4ht7eIbpJZnCqg9STXll98UdX8U3kmk/DjSHvWB2y6rdIUgiz3AOM+27nj7pqGy8AeI/H90mr/EK7eCxzvg0O2bYiD/AGyDwcZ/2ueoxipfEPi5orxfh/8ADWygOocpPcQIFhslP3jkcbhnk84Pq3AAOF8R6GtnrUVjfXU/jLx5dfKsG4m2ss4wWB+9gngHagHLKBw3b6X8CdJPhi5g1qX7Rrl0Az3sf3bdx0EakAFexBHPtxjrvAvw+0/wXZO6ubvVbn5ry/l+Z5WPJAz0XP4nqTXX8/3cYoA+VdM8BWmm+KZPC3iK/utD1gn/AIl+pQtutrkE8Ag4OT2wwHYgHr3o8N/EDwngNoXhvxVYIcfLaxxzhB26Lyf+BnJ716V408F6X420RtP1CMpIuWt7lFBeB/Ueo9R0I9Dg1xHgvxpqfhnXF8DeOXC3a/Lp+pMx2XK9FUse/YN+B5HIBWsPiV4FS4Wy8S+Ej4evcAsl3pqso/ELu/EqK9B0m08Ga3befpVpod5D3NvDE4X64HB9jWxfabYapbG3v7O3u7c9Yp4ldfyINcDqnwS8MXFwLvRnvdCvVJZJrGdgFY98HOB7KRQB2v8AwjOg/wDQE03/AMBI/wD4mj/hGdB/6Amm/wDgJH/8TXnR0r4u+E+dP1Wx8UWagARXi+XNjuckjn6ufpUtv8a7bTp1tfGHhzVdBuGO3e0RkiOO4OAT+AP1oA9A/wCEZ0H/AKAmm/8AgJH/APE0f8IzoP8A0BNN/wDASP8A+Jpmh+KtA8Rpu0fVrS8OMlI5BvA90PzD8RWzmgDJ/wCEZ0H/AKAmm/8AgJH/APE0f8IzoP8A0BNN/wDASP8A+JrWooAyf+EZ0H/oCab/AOAkf/xNH/CM6D/0BNN/8BI//ia1qKAMn/hGdB/6Amm/+Akf/wATR/wjOg/9ATTf/ASP/wCJrWooAyf+EZ0H/oCab/4CR/8AxNH/AAjOg/8AQE03/wABI/8A4mtajOKAMn/hGdByP+JJpv8A4Cx/4V5n8Vl08LYeDPD2j6euu60wXdHboDDDn5mJAyM4POOiv3Fepa5q9noOjXeq38my1tYzJIe5x0A9SegHcmvOPhVpF5rl/qHxE1yPF7qjFLGJukNuDjj0zgAHHQZ/ioA858V/DiHwL410T7Pq13p+nXyLBFqkbYa2uQoBLYx8pOD1HBbH3a7uDx14w+H8qWvj7Tvt+mZCprViu4DJ43gY/UKev3q9A8aeF7bxj4Wu9HuMK0o3QSsM+XKB8rY/mO4yO9cx8K/EcniDw7c+HNdhB1fR2+x3kM43eZHyqk54bgFT6kZ7igDuNF1vTPEGnrf6Tew3dq/R4z0PoR1B9jzWhkeoryjWfhZe6LqL698O7/8Asq++9Np7N/o849Mdu/B4yeCuM1qeD/ihBq2o/wBg+JLRtE8RIQhtpsrHM3T92T6kcDnqMFuaAPRKKRenbFLQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFNY9DnimXE0UEDzyyJHFGpZ3ZsBVA5JPYD1rxvVfEOtfFvVJ/D/hOV7Lw1Edl/q5Qjzh3RRxkEH7vBPcgdQDR8SfEPU9e1h/Cvw8jS6vx8t3qZ5gtV7kNjBI9eRxgBj03vBPw10zwhv1C4lfUtcnJa41C5O5iT97bnOB1ycknnJ7VteH/DujeCdAFlp8UdtawKXmmkYBnwOXkf1xyTxjsAOB5tqmv618XNTn8P+FZZLHwzE2y+1bYQZvVE6cEH7vUjrgHBALfiLxrq3jjWpPCXgGQiJONQ1pD8kSnqI29evI5PO3puruPBvgnSfBOjix02LMjHdcXLj95M3qSOg9FHA+uau+GvDWl+FNGi0vSbbyLdMlieWdj1Zj3JwOfywBitkUAIO9LRRQAjdq5rxp4L0vxtojafqEZSRctb3KKC8D+o9R6joR6HBrpqRu1AHkfgvxpqnhjXF8DeOZNt2pC6fqTMdlyn8Klj39CevQ89fW0wFrm/GngvS/G2iNp+oRlJFy1vcooLwP6j1HqOhHocGuI8F+NNU8Ma4vgbxzJtu1IXT9SZjsuU/hUse/oT16HnqAeuGobm0t7yB4Lm3jnhcYaOVAysPQg9alTAWnUAeea38GPB2rS+fbWUulXQIZZdOk8rBHT5MFR+AB96x/8AhFfij4UXOgeJ4NftVyRaaonzn0AYkn/x9RXrdFAHky/GG+0GVLfxt4R1HSeQn2qAebCzd8HgY9gzV2/h/wAdeGPEwA0nWrWeVhnyS2yX/vhsN+lb7orqVZQykYYEZBHpXD6/8IPBevBmfR47OcjAlsT5JB9do+Qn3KmgDu6K8l/4QHx/4YYN4U8Zte2qnIstXXcMD+ENhv02U3/hafijw0pXxt4IvIIkGWvdP/eRD06kqP8AvvNAHrlFcjoHxL8I+JGWPT9btxO/AguD5UhPoFbGfwzXWjGOMYoAWkNLmuT+InjCHwX4RudT+V7ph5NpEed8pHGfYYJPsKAOJ8czy/ELx/Z+ArGRxpdiwutXmjPp0jB9cHHszZ/gr122t4rS2jt4IlihiUJGiDAVQMAAegFcR8LfB8vhfw49zqQZ9b1R/tV/I/LhjyEJ74zz/tFu2K7taAA15D8SLO48EeL9P+I2mRM0BItNYhTjzI2wA/HsAPqqe9ev1T1XTbXWNLudOvYhLbXMZikQ9wePwPv2oAfZXdvf2cN5ayrNbXEayRSKeHU8gj8MVz/jLwJovjbTjb6jBtuFz5N5EAJYjx0Pcccg/wD1xx3wt1K68Na5qPw41mVjNYs02myvwJoDzgeuM5wOnzD+GvWRQB45p3i3xD8M9Sh0Px0z32jStsstbQFiP9mTv05PUjBxuHI9etbiG6to7mCZJYJVDxyIQVZSOCCOoIxUGraVY63ps2nalbJc2c67ZInHBH4cg9weoIyMGvHimt/BO/DJ9p1TwRcSYZT80tizHr06E/gx4OCRkA9uoqlpWqWOtabDqGnXUdzaTrujljOQR/MEdCD0PFXaACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooozQAVHM6RRNLIyqiAszOcBQByT+FOcZGO3868c8V6xf/E3xLJ4J8NTtHo1swOs6inQjP+rU9+QQB/ER/dBNAFfU9S1T4ya9NoeiSyWnhCzkH26/AwbojB2r/QenzH+Fa9StrfRfBnhsQxiDTtKsUyWY4VB6knqSfqST6mqxfw/8O/CKB2jsdLsY9ozyzH8OWZjn3JNeb2emaz8ZtUi1TWY59N8GW77rWwBKvd4/iYj/ANC7che7UAEk2s/GzUmt7b7Rpfge3lxLJwJL1ge3HT25A6nJwB69pGkWGhaZDp2mWsdtaQLtSNBgfUnqSepJ5J5qeytYbG0itbaFIIIUEccUa7VRQMAADoKnoAKKKKACiiigAooooARu1c1408F6X420RtP1CMpIuWt7lFBeB/Ueo9R0I9Dg101I3agDyPwX401Twxri+BvHMm27UhdP1JmOy5T+FSx7+hPXoeevraYC1zfjTwXpfjbRG0/UIyki5a3uUUF4H9R6j1HQj0ODXEeC/GmqeGNcXwN45k23akLp+pMx2XKfwqWPf0J69Dz1APXaKamAtOoAKKKKACkIz2paKAOQ174Z+EPEW5r7Q7YTNyZoAYXz6kpjP45Fcl/wq3xV4ZBPgnxtdQwqPksdRUSRD15wVH/fGfevXKDQB5GfH/j7wwwTxZ4Le6tgTuvdKbeAo/iKjP6la5nT/GGh/Eb4q22p6vqEOn6Lo8YfT7W+kSMyzEjk5O3OQDwf4VHrXcfFzxLd2un2nhPRMvrmuv8AZ0CNgxxHhmPpnpk8Abjn5au2Xwj8Ip4YstIvtGtrl4Iwsl1s2TO3Unep3dSeM4FAHdRsrIGQgqQMEHIIp9eSt8HtS0AmXwT4v1LSyCXFrcN5sJJ9hgfmrGkPir4o+FSw17wrBrtomAbvSmO8ju20An/xxRQB63SHmvPNF+NHg3V5RBPfSaXc5IaHUY/K2kerZKj8T+Fd9b3MF1Cs9vPHNE4yskbhlYeoI4oA83+Lnhy7eys/GOiDZrWhN5wI/wCWkAOWU46gdcehYd67Xwr4hs/FXhyz1qyIEVygZkzkxv0ZT7g5H4ZrXdVddrAEHggivHfDrN8MfihceF5iU8Pa8/n6a7fdim4BjHv0X/v360AeyVDdW8V3bSW88SSwyqUkjkXcrKRggg9QQelSIMZzTqAPELyx1X4Ja42p6XHPfeCr2UfarQHc9m5OAQT+jH7w+VsHax9k0zUrPV9Og1DT7hLi0uEDxSoeGB/kexHUHrUl3bQ3lrLa3MKzQTI0ckbgFWUjBBB7EZrxlTdfBPxQI5DNN4G1Sb5ScsbKUj8+g+pA7leQD22ioreSOaFJopFkjkUMkikEMCMgg+h61LnPSgAooooAKKKKACiiigAooooAKbJIkUbSSOqIoLMzHAAHcmnVHNGksbRyIrowwVZcg/h3oAht9SsbyQx217bzuBuKxSqxx64B9xRcanYWknl3N7bQyYztklVTj6E1892AXwlr+gfESAeVpmsald294ijCRxPM4jOB22jd/wAAFeo+OrCz1XxJ4O02a1gkkn1IzuzxhiY4YncryPuklf0oA7qGaK4iWWGVJI2+66MCD9CKce1RwRRwRLFFGscajCoi7VH0Fc9488X2ngrwtcarcBXl/wBXbQn/AJaynoPpwSfYHvigDlviZ4s1CS7tvA3hc7te1MbZZVP/AB6wnqxI6EjJz1A56kVoWy+Gvgz4EVZ5flXLMwA868mxzgev6KByeM1x+i3Nr8MfD9x4t8Vu934u10l0tePNOeRGP7o5BY9BkKASADq+FPAuqeJtZj8Z+PR5l0cPY6WRiO1XOV3L69MKeR1bJ4ABU0Hwrq/xO1eHxV40ia30eM79M0bPysh6O/qDx15b2XAPsMMaxRrGiKiKAqqowFA6AU5BjOT3p1ABRRRQAUUUUAFFFFABRRRQAUUUUAI3aua8aeC9L8baI2n6hGUkXLW9yigvA/qPUeo6Eehwa6akbtQB5H4L8aap4Y1xfA3jmTbdqQun6kzHZcp/CpY9/Qnr0PPX1tMBa5vxp4L0vxtojaffxlZFy1vcoAXgf1HqPUdCPQ4NcR4L8aap4Y1xfA3jl9t2pC6fqTMdl0v8Klj39CevQ89QD12impgDFOoAKKKKACqeq6laaPpdzqV9KsVraxmWVz2Uc/ifQdzVpu1eQ/EK6uPHPjKx+HemSslrGVu9YnQ/cQYIT69D9WX0NAE3wu0278T69qPxH1mJlluyYNMhbnyYAcZH8vwY/wAVesKMdsVDZWlvYWUNnaxLDbwII4o1GAqgYAH0FWKACkIzilooAxNb8J6D4ji2axpFpd8Eb3jG9c+j8MOnY1wlz8FIdOuHuvB/iTVNAmY58tJDJEfYjIY/iWr1aigDyP8AtX4ueElzqWkWPiizQEtNZHZPjsMADP4IfrXPeNvHnhXx34cl0u/S90DxBZkT2f26ErtlA+7vH3QemW2gcHtXvhrO1bQtJ1uHydU061vE5wJ4VfGe4yODQBznwv8AGaeNfCEV3K6/2jbHyL1O+8dGx6MOfTOQOldpkEZyK+b/AIh+A4vA3ivS9U0W/vNE0W+YW8txbSMWs5PXO4EqfvYz/C3tXbKfi74WYFf7O8WafkEEYin2Y/Dn/vugD1vNZ+taRZa9pFzpeoQrNa3KFHQnH0IPYg4IPbFee6f8b9FjuFs/E+l6p4fvf4kuoWZB+IAb/wAdAr0DStd0nXIPO0nUrW9jHVoJVfb9QDx+OKAPL/A2q33gDxSfh94hnMllM27Rb5+FkUniMnsSTwOzcdCtewrwK5D4h+CYPGvhxrVdsWo2582xuM4Mcg7ZHO09D+B7Cqfwu8ZzeJtFmsdVzH4g0t/s9/E4wzEEjfj3IIPuD0BFAHeUZopD2oAXNFeF/EbxNrvhXx9bTafq98dJt5LaXUIXkDIPNkkO0Dsu2Mj8a9vd0jjaR2VY0BLEnAUDvQBLkUV4RD4t8Q+LvjDYada6leWOg3UTTQxWxCs0CB8SEkZG8rkf7LLXuNtD9nt44vMeTYoXfI2WYDuT60ATUUUUAFZHirUf7I8J6vqAOHtrOWVP94ISP1rXrm/Gvh2/8VaHJpFrqdvY28423BktDMzjIIC4kUKODnIOc9qAMG78Frq3wUtvDQQfaF02Ixdv36qGH0ywP4E1zHwx8Qv4x17Q5Ljc0+haLJDOWByJ2kVAT7mOPP8AwI167YRXsNoqX08E9wOskEBiQ+mFLMR+Z/Cuf8MeB7Twv4g8Q6pbTBhrE6y+V5e3ycbiQDnkFmJ6DFAHVOQFJJwO9fMnjD4hWWs+OP7cniW80rSWaDSbU/6u5uBjMr/7AO1iM5OEHdiPVPi/4kubLRrXw1o7bta16QW0SqSGWMkBmz2zkLk46k9qfd/Bvw3qHg7TdBuRMsmnxsIruFirb3wZG2nKncRnkHAwBigDlPBMfhpNVPi7xn4t0W+8RzYdIzexFLIdgoDYLD24HOMnk+mDx/4OBOfFOjf+Bsf+NeNzeA9W+Hspe/8ACmm+LtCB+aaG2CXMQ9SAM+5JDdOq12fhG3+E3jSIf2Zo+lfasbmtJ4BHMo7/AC85A9VJHvQB2X/CwfBv/Q1aN/4Gx/40f8LB8G/9DVo3/gbH/jVf/hWngj/oWNM/78ij/hWfgk/8yxpn/fkUAWP+Fg+Df+hq0b/wNj/xo/4WD4N/6GrRv/A2P/GoP+FZeCf+hY03/vyKP+FZeCf+hY03/vyKAJ/+Fg+Df+hq0b/wNj/xo/4WD4N/6GrRv/A2P/GoP+FZeCf+hY03/vyKP+FZeCf+hY03/vyKAJ/+Fg+Df+hq0b/wNj/xo/4WD4N/6GrRv/A2P/GoP+FZeCf+hY03/vyKP+FZeCf+hY03/vyKAJ/+Fg+Df+hq0b/wNj/xo/4WD4N/6GrRv/A2P/GoP+FZeCf+hY03/vyKP+FZeCf+hY03/vyKAJ/+Fg+Df+hq0b/wNj/xo/4WD4N/6GrRv/A2P/GoP+FZeCf+hY03/vyKP+FZeCf+hY03/vyKAJ/+Fg+Df+hq0b/wNj/xo/4WD4N/6GrRv/A2P/GoP+FZeCf+hY03/vyKP+FZeCf+hY03/vyKAJj8QPB3bxTo3/gbH/jXNeNLv4deNtDbT9Q8TaMsi5a3uUvIy8D+o55HqvQj0ODW/wD8Ky8E/wDQsab/AN+RR/wrLwT/ANCxpv8A35FAHnnw/wDinFpV7J4T8V6vZTm2JW01iO5V4ZlA4Dvng46MfoeRz6QvxA8Ggf8AI06MAf8Ap8j/AMax9e+EHg7VtIntINHt7Cd1/d3Nqm1427H0I9Qevt1rznwjpvh/w/4h/wCEK8d+HdMF9uxYam0ICXak4UE46+h79DzyQD2D/hYPg3/oatG/8DY/8aP+FgeDf+hp0f8A8DY/8arj4aeCT18M6af+2Io/4Vp4JH/MsaZ/35FAFDxP8VPDOjeHry+sNa07ULyOM+RawXCuzueBkA5wDyfYGuW+FOpeGPD2hTaprHijSG1/WJPtV40l5HvXJyEPPHUkjsWI7CsXRvCOgeOvijcyaZpFrD4W0P8AdOYUwt7Nk9x1Xj8lH96vUv8AhWfgo9fDGm/9+RQBOvxA8Ggf8jTo/wD4Gx/40v8AwsHwb/0NWjf+Bsf+NQf8Ky8E/wDQsab/AN+RR/wrLwT/ANCxpv8A35FAE/8AwsHwb/0NWjf+Bsf+NH/CwfBv/Q1aN/4Gx/41B/wrLwT/ANCxpv8A35FH/CsvBP8A0LGm/wDfkUAT/wDCwfBv/Q1aN/4Gx/40f8LB8G/9DVo3/gbH/jUH/CsvBP8A0LGm/wDfkUf8Ky8E/wDQsab/AN+RQBP/AMLB8G/9DVo3/gbH/jR/wsHwb/0NWjf+Bsf+NQf8Ky8E/wDQsab/AN+RR/wrLwT/ANCxpv8A35FAGd4o1/wH4q8OXujXnijRvKuY8BxeR5jbqrj5uoOD+HpXMfCn4kaZBoEugeItZsILvSZDbxXElwojuIhkKVYkBsYx7jae9dufhn4KHI8Mab/35FecfFH4f6V4aGm+K9G0O1a1sZQuoWOz93LETjdjsecZ9we1AHol/wCLvAGqWxttR13w9dwHrFcXETr+TGuA1Xwn8Jrq5+2aR4rs9DvVJKzafqiKA3+6W4H+6RXb6X4F+H+s6Va6lZeHNMktbmJZYn8kcqRnn39RVv8A4Vn4J/6FjTP+/IoA8s/4SrxX4UT/AIl3xA8M+KLRMBYr27jjmPryXBP4uelcnffFKCP4h2Xi2x01rG/UeRqcEU6yw3cfAyGGOcY45Hyqc8GvoD/hWXgn/oWNN/78imS/DDwS8Tx/8I1p6h1KkrHtPPoR0oA29C1zTPEelx6lpN0tzaycB1BGCOoIIyCPQ1pE+nJrwn4JXdp4d8a+K/BrzlpEumFs78GURM6N+OMNj0B9K9sv55bazkmgsp72Rfu28BQO+eOC7KvqeTQB474s0weIfDHxO1JV3NHeRRR98C0RC2P++pPzNbkesy+NfDHhrw7ayt5mqWMdxqsyH/VW6/LIM+ruGQe249q1PBukak3hHUdH13RriynvXupbh5JYZElM7uSBsckkKwB3AfjVf4e+E7r4feCl8zTp9Q1m4Ia5jt5I9wHO1Azuq7VBPfqTjNAGF4Lgj1D49eKb2CNVtNLtUsIVUcRkbVwP+/b17CK8u+FuheINC1bxBc67oVzbz6veG5Nx58EiAZYgNtkLZy56KfrXqIoAWiiigAooooAKZKyohdmCqoJLE4AHrmn1518ZvEE2leCv7MsSW1LWpRY28aEbiG++QD2x8vsXFAGL8Pkbxz8Qta8eXClrK1c2GkgjhUA5cA9DtP5yuOwr15axfCOgQ+FvCun6LBgi1iCuw/jc/M7fixJ/GtugBGGeK4fxZ8LPDviqY33kyadqwO5b+zOyTcOhYdG5wcnnj7wruaKAPHP7a+Inw3bZrlq3ijQF/wCX6AH7RGvq4PPQE/Nkcj5677wp468O+MoDJo9+jzBcvbSfJMg91PbnGRke9dIRmuA8U/CXQfEF1/aNj5mjaup3pe2HyHf2LKOD1zxgn+9QB6BmjIPQ147/AMJR49+HDeX4r09vEGhp01SyH72NR/fX2/2sdfvGvQ/C/jHQfF1oZ9G1GK42gF4vuyx5/vKeRznnGD2oA36KMj1ooAKKKKACiiigAooooAKKKKACiiigBG7VzXjTwXpfjbRG0/UIyki5a3uUUF4H9R6j1HQj0ODXTUjdqAPI/BfjTVPDOuL4G8cSbbtSF07UmY7LpeiqWPU+hPXoeeut8WPFNxpGhweH9HzJrutv9mto04ZEPys3t1wM9yT/AAmtn4g+F9C8S+FrlNbcW8VtG00d6B81vgZLD1HHK9x6cEeKfCbXbeTx7Z6j4wurt72azFvo11djEZAJQgN68FR2JLZOSKAPePBPha28HeFbTR7chmjG6eUDHmyn7zf4ewAroaauBn+dOoAKKKKACiiigAooooAKKKKACq99Z2+oWU1ndwrNbTo0UsbdGVhgg+2KsUh6igDx/wCHV3ceCfGGofDnU5Wa3Ytd6PK55kjYklfrwT7FX9RXr69688+LPhW51jQYNc0ndHruiN9qtXQfMyghmX36ZA9Rjua6PwR4ptvGPhWz1m3wrSrtniBz5Uo4Zfz6Z6gg96AOipDwKWkNAHy/r+malY/EHxl4p0w/6ZoGpQXRjXo0Um7dnHX+HPsWr6S0XVbbXNGs9Vs33W93CsqZ6gEZwfcdD7ivM/Ddra3Pxp+IWmXUPmQ3drAHjYcOhjXcD9d1O+EF1PoWo+IPAF/IXm0i4aa0LYy8DnOQB0GSrf8AbSgD1mikFLQAUUUUAFFFFABRRRQAjdq8jAPjP4+kMd+m+FrfgdUa5fv7HP8A6Jr0/WNSh0bRr3U7j/U2kDzuB1IVScD34rz/AOCGmzJ4OudfvcNf65eSXcsm3BK5IA+md7D/AHqAPTV6f54paBRQAUUUUAFFFFADW57ZHcV554l+EOi6vef2po80+gaypLLd2BKKW7kqCPzUqeec16LRQB46vjbxv8PW8jxvpTarpanC6xp65Kj/AG14B7DnaeD96vSPDninRPFNkbrRtRhuoxjcqkh0/wB5Tgjp3Fa8i7l2kAqRyCM5HpXm+v8Awd0m6vf7V8NXU/h3V1JImsyRGx75QYxn/ZI+hoA9LyKK8fg+IPi3wI4tPiDoz3NmDtTWdPTch56uowO/+yf9k16XoPiHSPElgL7R7+G7tzxmM8r7MDyD7HFAGrRRRQAUUUUAFFFFABRRRQAUhpa5/wAaeKLXwf4Xu9YutrGJdsMRbBlkP3VHf3PsCe1AHBfE2/ufGHiaw+G2ky7RMy3OrTLz5UK4YKfTs2DjJ2evPW+I/h5ofiLwlDoElsLeK1jC2U0YBe2IAAI9c45B6885wayPhL4WuNO0i48SayC+u6632md2GCkbHcq47dckdsgfw16Kc+lAHkngvxpqnhjXF8DeOZNt2pC6fqTMdlyn8Klj39CevQ89fW0wFrm/GngvS/G2iNp+oRlJFy1vcooLwP6j1HqOhHocGuI8F+NNU8Ma4vgbxzJtu1IXT9SZjsuU/hUse/oT16HnqAeu0U1MBadQAUUUUAFFFFABRRRQAUUUUAI3bgmvGrTHwr+K5sj8nhrxK+6EYwtvcZ+6PQZYD6Mv9017NXLfEHwjD418I3WlPhbgfvbWQ/wTAHafoeQfYmgDqF7/AFoNcH8KfF0viTwy1nqRKa3pL/ZL6Nz85YZAc+5wc/7StXeE9PSgDyjQRJD+0p4ojJHl3GkxS/8AfIgX/GoviOR4Q+JHhjxsg2Wsz/2dqDE4Xac7WPrgFj/2zFWbcC3/AGk7k7WDT6ECT64def8Ax2uk+J+gjxF8PNXslQvOkJuIAoy2+P5gB7nBX/gVAHXgd/wpa5L4aa8fEfw90fUJHLz+QIZ2Y5JdCVJP1xn8a62gAooooAKKKKACiikJxQB5r8ctSltfh/8A2bagtc6tdxWcaqeTk7jj67QP+BV3eh6bFo2hWGlw/wCrs7eOBT6hVAz+OM15t46K658afBGgkMY7MSalIO3GSufxhx/wL3r1gUALRRRQAUUUUAFFFFABRRRQAUUUUARzRrMhjkRXjYEMrDII9CO9ea638HrJb46v4O1Cfw7qo5H2cnyH74KDoDxwPl/2TXp1FAHj9t8TPEfgyWOw+ImhypETtTV7JA0T+5A4zwTxg/7NeoaNrWma9p632k3sN3atwJImzg+hHUH2ODVq5tobu3e3uYY5oJFKvHIoZWHoQeCPrXmOr/CBbK9fV/Auq3Hh/UuvkqxNvJ32kdhk9MMv+zQB6pkUV5HZ/FLWPCtzHp3xG0KWxY/ImqWqb4JfcgZ+p2knn7or1DS9UsdYsUvdOu4bq2k+7LE4ZT7cdD7UAXKKKKACiiigBDXjVyR8Vvip9jA8zwx4bffNgfLc3HTb7jII+in+9XT/ABX8Xz+HfD0Wn6VufXNXf7LZJH99c4DOPQjIAPqwPY1seAfCMHgrwna6THtaf/W3Uo/5aSn7x+gwAPYCgDpVzzkGnUUUAI3aua8aeC9L8baI2n6hGUkXLW9yigvA/qPUeo6Eehwa6akbtQB5H4L8aap4Y1xfA3jmTbdqQun6kzHZcp/CpY9/Qnr0PPX1tMBa5vxp4L0vxtojafqEZSRctb3KKC8D+o9R6joR6HBriPBfjTVPDGuL4G8cybbtSF0/UmY7LlP4VLHv6E9eh56gHrtFNTAWnUAFFFFABRRRQAUUUUAFIaWigDx7xxDN8PfH9l48s0b+y74i01iNBwM4xIB64GfquP4q9chmiuoEngkWSKVQ6OhyGUjIIPpg1T17RrPxDod3pN+m+2uoyjjuPQj3BwR7ivOvhLrF5pV1qHw+1tv+Jjo7E2sjZ/fWxPGM+mQR7MBgYoAXUZDD+0rpIBGJ9DZT/wB9yn/2WvU2wRzXlPiZBH+0R4MnKHc9jPGWHfCTYH4bv1r1b9cUAeUfB/Og694w8HMNsdhqH2i2BOSY3BA/8dWM/Vq9ZFeS3n/Ei/aOsJ1TbDr2ltFI/YugJ/PEcY/4FXrI70ALRRRQAUUUUAFIfrS0hoA8p8Oq2q/tCeJ74kNDpljFaR8fdZghP6iT869WHSvKvhDGbnxF491gkt9p1holJ7BGcgfk4r1agAooooAKKKKACiiigAooooAKKKKACiiigAooooAr3tlbahayWl3bxXFvKNskUyB0Ye4PBry/UvhDc6Pdyap8PtZn0W9PJtJHMlvL7HOfyYMPpXrFBoA8m0/4s3mg3UelfEPRJtIuT8q38SF7eY+oxnHUdCevOOlen2F/aalZx3lldRXNtKMxyxOGVh7EcU3UtMsdXspLPULSG7tpBh4pkDKfwPf3rzC/+FGqeG7uTVPh3rUumzMcyadcOXgl9snPv97J56rQB63moLu4gtLSW5uZEighQySO5wEUckn6AZrzHS/i62nXqaT490efQL9vlW42l7aU8ZIPOBk/7QHdhVX4ka8/i7UNL8BeG7tJTqZWe+uYH3rFbj5sbhxz1/75H8VADfh/bXHj7xte/ELU4WFjAxtdHhccKoyC+PUZP/Amb+7XsIqjpOmWmiaVa6ZYxiK1tYxFGnoB6nuT1J71eoAKKKKACiiigBG7VzXjTwXpfjbRG0/UIyki5a3uUUF4H9R6j1HQj0ODXTUjdqAPI/BfjTVPDOur4F8cPi8UhdP1EsSl0ucKpY9T6E9eh56+tpgLXN+NPBWl+N9EfT9QQpIuWt7hR88DY6j29R39uo8A8SfEvx54Tik8G3GoxC809zFJfx4eWWMgFAWPfBzn73IB5ByAfUtFfPvwh+Lesal4jg8PeIboXUdypW2uHUB1cDIUkdQcEZPOcetfQK9O9AC0UUUAFFFFABRRRQAh6V5Z8WtDu7GWw8faIh/tPRWBuEA4mt/4g3qACQf9lm9K9UqK4hjuIHhmjWSKRSjowyGBHII7gigDyDWtattc+I/wx16xkBtr1LjZk8qSoBU+4LEfUV7H1P0r5iu9Cu/BHxl8OaGzO+kDVVn04nnCzuisM+20Aj2z/FX06KAPKfjCv9naz4H8QhiostWWFyO6PhiPyjYfjXq9eZ/HmyW6+F9zORk2lzDMvsS2z+TmvQdMuVvdMtbtTlZ4UkB9Qyg/1oAt0UUUAFFFFABTJWCoWPAXknGcYp9VtQkaHTbqVRlkhdhn1ANAHmX7PyM3gG8u5CC91qc0pwMY+VBj8wfzr1avN/gTGU+FOnsQAHmnYY7/ALxhz+VekUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUhpaRu1AHL/EC60PT/AAXqF5r9lBeWUKZFvMoPmSdEVeCQSTjI5GSe1eOeCvht4z0jQ7Xxd4dvre11S5Qt/Zs8fyvbkhlUuScFsA4OOo+YHNdPrOfil8UI9Bi/eeGvD7+dfMM7Zp8kbP0K/TzD6V7Eq7QABgDgADGBQB5ho3xht4L9dG8baZP4e1QHbumUm3k7bg3VR19V/wBqvTYJoriFJ4ZEkjkUMrowIYHoQR1rP1vw/pfiKxNnq+nwXkHZZVztPqp6qfcEV5rP8OvFXgedr34eaw0lpku+jag+6NvZScDPT+6f9o0Aev5FFeZaD8X9Olv10fxTYz+G9XHymO7yIW7ZDnGM46kY9Ca9KiZXQOjKyNypU8Ee1AD6KMj1ooAKKKKAEPavD/i78I9U8Sa03iDw+kMk8kQW5tWfYzuuAHUn5T8oAIO37o65r3GqEus6XHIY31K0V1OGUzoCD6EZoA+bfhn4fh8IfFS1h8YxPp18Ii1gkwUxSyMMDLgkZwTj/a7ggA/T6cDGRXMeMPB+keO9A+x3YBJG+1vIsM0LHkMp7g9x0I9Dg1xfgvxpqnhjXF8DeOZNt2pC6fqTMdlyn8Klj39CevQ89QD12impgLTqACiiigAooooAKQ57daWigDxb9oK0uGtfDN1Y7hfJfGKApw+5gpXae3zIK734d+L4vGnhG21LKreJ+5u4h/DKOpx2ByGHscVzHxwZYNJ8M3RTd5WuwHj02scfjt/SsvUAfhd8Vl1Jf3fhjxIxS64+S3ueu70A5z9Gf+7QB2nxYtlu/hdr8TchbbzfxRlf/wBlrQ+H8vnfDvw4+cn+zbdSc56RgUzx+Afhz4kJ5H9m3B6f7DYqr8LJfO+GHh9tu3FqFxn+6SP6UAdhRRRQAUUUUAFVNU/5BF7/ANcH/wDQTVuo7iMS28kZGQ6lSPXigDz74GsrfCbSgDkq84PsfNevRa8w+AUwk+F8CAY8q6mU+/Ib/wBmr0+gAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKM460AFcN8UvGEnhXwx5enln1rUX+y2EcYy284BYDvgEY/2io712s80VtBJPPIscUal3dzgKo5JP4V5D4Jgm+InxCvPHd4jf2Rp5Nro8TjAYj/lpj6En6tj+CgDtfhz4Oj8FeEoLBtr30v769lHO+U9eT2HAH0zjJNdbSDv1/GloAKQ0tFAGPr/hnRvE9mbXWdOhvIRnbvX5k9drDlT7g5rzeTwL4z8ASm58B6q2o6aMltH1Bs4H+weB6ngqeB96vYKQ0Aec+G/i9pGo3g0rX7abw9rA4a3vflQn2c4/UDrxmvRQykZBBB7+tYniPwhofiy0+z6zpsNzgfJJjEif7rDkdvyrzr/hD/HXw6cy+DtROtaMDk6Vfn94gz/AeAeM9Cv+6aAPYs0V574W+Leha7cjTtTSbQ9ZU7Xs7/5Mt6Kxxn6EA+1egjj2oAG7V4xqGnWOs/tJWdqLS3aHTdNM1xH5QIdyGILccn96nX0r2cmvIvhwP7X+L/j3XD8wglWyjb1AYrx+ES0AetIixoqRoFRQAqqAAo9PpXO+NPBel+NtEbT9QjKSLlre5RQXgf1HqPUdCPQ4NdMKRu1AHkfgvxpqnhjXF8DeOZNt2pC6fqTMdlyn8Klj39CevQ89fW0wFrm/GngvS/G2iNp+oRlJFy1vcooLwP6j1HqOhHocGuI8F+NNU8Ma4vgbxzJtu1IXT9SZjsuU/hUse/oT16HnqAeu0U1MBadQAUUUUAFFFFAHlH7QasPh9aTo+x7fU4pVPvskA/nn8K7bxj4YtPGPhW70e52/vV3QS4z5cg+4w/Hr6gkd65X48ReZ8LL1im7yp4Xz6fOB/XH416FZyedY20mMb41OPqAcUAeP6F4putS+EPizw/rOV1zQ9PuLa4WQ5aRBGwV/foQT7A/xV2nwk/5JXoH/AFwP/obVwXxt0C70WWXxlpC7Rd2z6dqqgcOki7Fc/oPqsfvXpPw2Qx/Dbw6CQc2ER491z/WgDqaM0hrjvHXjr/hBbGK/utKlubF3EXmRzKp8w7iF2nnop5oA7LNFcJqHj3U9F09tQ1fwXq1vp8Y3SzwzwTFAe5UPnH4cV1eiazY+INIt9V02YTWdwu6N8Y74IPuCCD7g0AaFIaWkNAHlfwMjWx0jxLpXAaz1uZSB2G1V/L5DXqteUfDpRpXxX+IWks3MlxFeovs25if/ACIterCgBaKKKACiiigAooooAKKKKACiiigAooooAKKKKACmt2p1ZPiXXrPwz4fvNZvmxb2sZfaDy7dFUe5PA+tAHn/xX1i71e80/wCHmhyEahqzBryRf+WNvyTn64JI9Bj+KvRdB0a08PaJaaRYR7La1jCIO57lj6knJJ9Sa8++Evh+8uEvfHetqDq+uHfHx/q7f+EAHscKf91V969RFAC0UUUAFFFFABRRRQAUjDIwQD7GlooA5zxP4I8P+L4PL1jTY5pAMJcL8sqfRhz+B4+tcB/wjvxC+G5L+HLw+JdCTn+zro/v41HOEP0GPl9fuV7FTWGe1AHnmg/FPS/FlvJp1pdR6Hr4IT7NqcJcK+eQAGTf3GMg55K8Yq14D8B3/gmW+Da3b38N9OZ591i0cpc+j+aRj/gJ69a0vFfw/wDDvjKEjVtPU3G3C3cPyTJ6fN3Az0ORXCjTfiP8Ngx0yX/hK/D6fMLaYkXMKDsp69MdNw/2RQB7EBilrifCPxO8OeLH+zQ3DWWpZ2tYXgEcgb0XnDdOxz6gZrtegoAG7VzXjTwXpfjbRG0/UIyki5a3uUUF4H9R6j1HQj0ODXTUh7UAeR+C/GmqeGNcXwN45k23akLp+pMx2XKfwqWPf0J69Dz19bTAWub8aeC9L8baI2n6hGUkXLW9yigvA/qPUeo6Eehwa4jwX401Twxri+BvHMm27UhdP1JmOy5T+FSx7+hPXoeeoB67RTUwFpcjGc8UALRRXJ+KfiN4Y8Ihl1PUkN0OlpB+8mPflR936tgUAZPxuXd8I9a5+6YD/wCR4xWtb+LNE0LwVo+oazqttbRy2MMgLvlpMoD8qjlvwBryzxt4n8beN/BeqS2vhpdK8MpF5ss17/rplUhhtB/A5APThq6D4e/Cfw3L4f0rX9Uil1a9urOGZRetujiBQYVU6EAED5sjjjHSgDmviL8U7jxT4O1K20HQrv8AsRtsc+q3K7F++vyoOmTx3zyTgYzXq3wy0G48OfD/AEvT7qbzJxGZGwchN7Fwo9hux9c1y3xyRn8HaRodntibUdUgtkRRgBcNgADsDtr1SNQiBVGFAAA9BQA41478crkXGoeD9EMc08dxqPnywwoXdlUqvCjknDtivYicV47rdxBqf7SOiwyzRrb6TYNI29gB5hDkfj8yflQBb8e/EGOfw3daNDo+qWUuqRtai61W2NpbxBxhmLv3wTgAH8K7XwH4eTwt4K0zSEuUufJjLNOhyrs5Lkr/ALOW49qwPiv4g0OD4f6rYz3ltNdXkPk21srq7vISNpCjng857Y9cVqfC/Tb7SPhtollqKsl0kLMyN1QM7Mqn6KQPwoA66kPSlpDQB5PcBdE/aTtJAmE1vSTGXxwXTJ/PES/mK9YFeUfGYHR7/wAIeLVLKNM1MRzbB1jfDHPthGH/AAKvV1IKgqcgjg0ALRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQA1x0rx3xO7fE74lweE7c7tA0R/P1ORTxLKMgR/zX8X9K7D4m+Mh4O8KyT25D6ndn7PYxAZLSHjdjuFznpycDvS/DTwaPB3hWO3uT5mqXZ+0X8xOWaVh93PcL09zk96AOwjRY0CKoVV4AAxgU+gUUAFFFFABRRRQAUUUUAFFFFABRRRQAUjZpaKAOR8WfDjw34wVn1CwEd4Rhby3Plyr7kjhv+BA1xWPiT8NAdo/4S7QI+QCSLmFB27t/wChjj+GvY6QjkECgDkPCHxI8N+MgI9PvPJve9lc4SYY64GcMOO2cd8V14NcX4s+GHhzxaz3Fzam01H7y31odku7sW7P0HJ59CK5L7b8R/hshGow/wDCW+H48k3ERIuYk/2s5JA5PO/p94UAewntjn6VzPjTwXpfjbRG0+/jKyLlre5QAvA/qPUeq9CPQ4NVfD/xM8JeI9Pa7t9Yt7fy13zQ3kixPEO+Qx5HTkZFc7qfxltLm8bS/Bej3fiK/wAfeijKQp2ySRkgH2AP96gCp4M8aap4X1xfA3jmQJdr8un6k7Hy7lOiqWPf0Y/Q89dvxF8X/DWjXH2Kwkl1vVCdsdrpw35bnALjjt0GT7VzF38MfF/xAKXfjnWoLNIwxtrCxhVhCWHc/UDIy2f7wqD4b31t8PfEMvgzxJptnY6jK5+x6qkYAvFJ4Vn6/T8jg9QDS/s/4o+Peb66j8JaO/8Ay7wZNy69skYI7jqnuprqvC/ws8L+FWWe2sftd+Dua8vD5shPqM8KfoPzrs16dfpzTqAOW+IsXmfDfxGpOD/Z8zZ+ik/0pnw0mE3wz8OsGDYsY0yDn7oxj9MVd8cRNN4C8RRIMs+mXKgep8psVh/B2f7R8J9BfbtISRMf7srj+maAMLx9t1n4x+BNEyT9laTUZAOmB8y5/GEj8a9XFeT+F2/4SH49eKNXDB7fSLVNPi4ztc/e/wDHll/OvWBQAEZIrn28C+EpGLP4X0YsepNjFn89ua6GigDHsPCvh7SphPp2habaTDpJBaIjD8QM1rqCM0tFABRRRQByXxN0Q+IPh1rViilpRbmeIAZJeMhwB9duPxo+GWt/8JB8O9GvmcvMLcQSljk74/kJP1xn8a6tuRj1ryT4WH/hF/GnizwNJ8sMM/2+yXOcxNgHn2UxfjuoA9dopB1NLQAUUUUAFFFFABRRRQAUUUUAFFFFABTJHVIy7MAoBJJOAB9ac1eW/FrX7u6Nj4D0M51fXDslb/nlb87icdiAc9flV/agDN8Kq3xO+JNx4vuFZtB0Zvs+lIQQHk6+Zg+n3ue5QfwmvZF6dMVl+G9BtPDOgWej2K4t7WMICert1Zj7kkk/WtWgAooooAKKKKACiiigAooooAKKKKACiiigAopMj1FLmgAorlfFfxC8M+EExqmpRi5A4tYf3kx/4COn1OBXE/8ACUfEbx6NnhfRl8P6TJ/zEdQ/1jL2KgjuPRW/3hQB6Vr3iTRvDdp9q1jUrezi5x5jfM/sq9WPsAa84l+KXiDxbPJafD3w3Lcxg7W1S+XZCh7kDOM4PAJz/s1o6J8GdFtrz+0vEdzceI9UPLTXzEx59AhJyP8AeJ+lekQwx28SxQxrHEgCqiKAFA7ADtQB4rH8A5NYN3qPinxA8usXXzFrGBEiR8YyQVG8fQJRYXXjP4P2Ys77RbbWfDUZJ+2adEIpY19XUd8DJLA8n75r26kPNAHOeF/HHh/xhbedo+oRyyAAvbthZY/95Tz+IyPemeNPBmleNtEbT79Nsigtb3KAF4G9R6j1HQj0ODWL4p+Eeg6/Ob/T/M0TWFO5L2w+T5vVlGM9eoIPvXOnxb49+HQEfi/TP7d0VTtGq2I/eKv+2vHbH3gvOfmNAE3gzxpqnhjXF8DeOX23akLp+pMx2XSdFUse/oT16Hnr62mAtef6jF4O+Mfhp7e1vYp5EG+KWPie0c9yjYOCRgg8EdD0NYngzxpqnhjXF8DeOX23a4XT9SZjsukzhVLHv2BPX7p56gHpmvRCfw/qULHAktZUP4oRXn3wm1WDSfgbbajcN+5sY7uWTHXCyO2Pr/jXpkqb7d4wAdykbT0PGK+ZrHWJn+A2meGbMKb/AFnV2sok3YLKHVyeP9pkU+zUAepfA7T5ovBM+s3h3Xms3st3I5GCRnaPwyrEf71em1S0jTYNH0ez0y2GILSBIE+iqAPx4q7QAUUUUAFFFFABRRRQAhryP4po3hTxn4Z8fQhlhhm+w6gVBP7ps4OB14L/AIha9drF8W+H4fFPhbUNGmIAuoiqMf4HHKN+DAH8KANeN0kQOjBlYZBU5BHXNPrzn4NeIZdT8INo+oZj1XQ5DZXMTn5gqkhCR9AV+qGvRqACiiigAooooAKKKKACiiigAoJA60UjfWgDO17WbPw9od5q1++22tYzI+MZPoBnuSQB7kV538J9GvdVu9Q+IOuRf8THV2K2qMOIbcHjbnpnGB6qoP8AFVLxrK/xG+Ilp4HtHYaPpjC61eVDwWHIjyO+Dj6sT/BXr1vDHbwpDCixxxgKiKMBQOgA9MUASCloooAKKKKACiiigAooooAKKKOlABRRVe8u7axtnuru4it4IhueWWQIqj1JOAKAJ8j1pHICkkgAcnNeYar8ZrKa+Ol+DtKu/Eeo9AYEZYV5xktjJA9cY96ojwF438cYk8b6/wD2fp7c/wBlaZwCMdGPI/Pf+FAG94l+L3hnQrk2VpNJrGpk7FtNPHmEt0wWHAPsMn2rB+yfFHx/k3M8fhDR3/5ZRZa6ZSM8nhgevUp9DXfeHPBXh/wnCI9F0qG3bGGnI3St9XOW9eOnpiugA5oA4nwx8KvC3heRbmGyN7fg7jeXp82Td6gH5VPXkDPvXaqMZp1FABRRRQAUUUUAFNdQylSAQQQQRnI9KdRQB5x4l+D+japdDUtCml8P6wh3R3Vj8q7u5KAjBxnlSOvOa4LxlL4httH/ALH+JWjfb9PRv9F8QaaoLwMR95hwCDwCp2ZAPUgGvoSmSIskbI6hkYYZWGQR6GgDxL4XfFVPtMXhXX9QiuHBEdhqYYhbgdFR8gEN0xnrwDzyeW+CHh2bUvHk080jy6foBkaJWPyiZztBA7ZCluO6iuk+MPw28J6V4aufEVqp0q8R1VIoAPKndjgDYcbeMn5MAAE4OK7H4N+Ebzwn4MI1Bl+1ahL9raPb80QKgBWPc4GT6Eke9AHoYPrxRketYPizRZ9fsLTT45Wjt3u43uykrRsYVyxUFcHkhV4I4Jryrxz4esLP4heDfD2jtewLezF70C+nYyQhl4yzkjhZORQB7pRXFQ+ALPTPFOkanpIuIYrYzNcpJfTSB8x7VG12I4JJ4x0rtFGBQAtFFFABRRRQAUjUtFAHj3i4/wDCu/ipYeL13Jo2tD7FqePupJj5ZD+AB9fkf+9Xr6MCMggg85FY3i3w5a+LPDd5o138qXCYSTGfLccq34EDjv071x/wj8S3Vxpt14S1oGLW9Cb7O6sf9ZEOFYeuMYz6bTn5qAPS6KQUtABRRRQAUUUUAFFFFAATjrXI/EfxingvwlcX6FWvpf3FnEed8p6cegGWP0x3FdY3AyTgDv6V41oQb4pfFGXxC4LeG/DzeTp64+Wafrv/AJN+EfvQB13wv8HSeFfDG+/3PrGoN9pv5H5cseQpPtn8yfWu4FItOoAKKKKACiiigAoooyPWgAoorlvFXxA8MeEUI1bU4luQMraQ/vJm7j5R0z6tge9AHU5B71l654i0fw5Zi71jUbezhOdplfBf2VerH2AJrzT/AIS34iePMr4T0UaFpb9NS1IfO6+qqQffoG+orS0X4MaRBef2n4lu7nxHqjfelvXJjHtsJOR/vEj2FAFCX4qa74puHtPh94bnvFDbTqV8uyBffGR29WB/2TT7T4RX2v3CX3xB8Q3OsTAgrZQMYreM98Yx7fdCdO9eqW1vDawJBBDHDCgwkcaBVUdgAOBUtAGfpWj6dolmLTS7C3s7cf8ALOGMKCfU+p9zV8UtFABRRRQAUUUUAFFFFABRRRQAUUUUAFNf7uB1p1cb8S/GS+DPCkt1Dh9RuT5FjF1LSnvjuF6478DvQBx2tn/hY/xetdCjBk0Hw432i+K8rJP2Q+vI2468SV7EvTpiuM+GPg4+D/CccN1ltVuz9ovpWO5jIf4c99o4+uT3rtKAENeQ/wDIa/abPdNE0vnPTLL/APb/ANK9Wv3vY7Vm0+CCe4H3Y55jEh+rBHI/KvO/C3hHxZovjrWvEd7b6NOdWdQyx30oa3QHouYfm4A/u9OooA9NFFNXPcc06gAooooAKKKKACiiigBrDOK8o+KOj33h/WLP4jaDETdWA8vUYQcefb9MnHoDgnsMH+GvWajmiSaNopUDxuCrqy5DA8EEdxQBR0DWrHxFodpq2nyiS2ukDqe4PQqfcEEH3FaVeLadK/wd8cnSLp2Hg/WZC1pK5ytpMcZBPYdAc9tp7NXs6kUAOooooAKKKKACkJApao6vqtnouk3Wp30ojtbWMyyt6ADPHqewHckUAcB8XPEd0llaeDdDO/W9dbyAF/5ZQnhmPoDyM+gc/wANdp4V8O2nhTw5Z6NZD93bphnxgyOeWc+5JJx26DgV57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I7gnp1oA31wowOOOleRfHq6a+0rRPCtn8+oanfoUQc/Kvy8j/edfyPpV3xB4q8b/D7woLrU9M0/WI4CsX26C4eNsHhXliKcZOM4bGSBxVvwV4Wi1LUYfHeq6tDrWp3cQ+zSQJsgtYzkbY1PII5BJweWyM5JAPQLOBbWzhtkOVijWMfQDH9KnpFHU0tABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAFe8toL22ltbqJJreaNkkjdcqwPBBFc/4c8DeHfDN5NNo9g1q7FgQLiVl5A/hZiP07D0rpW+8P8AdP8ASmRf62T/AHj/ACFAEifxfXFOpqfxf7xp1ABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAH/2Q==']	Multimodal	Competition	False				Theorem proof	Geometry	Math	Chinese
21	"如图, $M, N$ 分别为锐角三角形 $\triangle A B C(\angle A<\angle B)$ 的外接圆 $\Gamma$ 上弧 $B C 、 A C$ 的中点. 过点 $C$ 作 $P C / / M N$ 交圆 $\Gamma$ 于 $P$ 点, $I$ 为 $\triangle A B C$ 的内心, 连接 $P I$ 并延长交圆 $\Gamma$ 于 $T$.

<image_1>
在弧 $A B$ (不含点 $C$ ) 上任取一点 $Q(Q \neq A, T, B)$, 记 $\triangle A Q C, \triangle Q C B$ 的内心分别为 $I_{1}, I_{2}$,

<image_2>

求证: $Q, I_{1}, I_{2}, T$ 四点共圆."	['连 $N I, M I$. 由于 $P C / / M N, P, C, M, N$ 共圆, 故 $P C M N$ 是等腰梯形. 因此 $N P=M C$, $P M=N C$.\n\n连 $A M ， C I$, 则 $A M$ 与 $C I$ 交于 $I$, 因为\n\n$\\angle M I C=\\angle M A C+\\angle A C I=\\angle M C B+\\angle B C I=\\angle M C I$,\n\n所以 $M C=M I$. 同理\n\n$N C=N I$.\n\n于是\n\n$N P=M I, \\quad P M=N I$.\n\n故四边形 MPNI 为平行四边形。因此 $S_{\\triangle P M T}=S_{\\triangle P N T}$ (同底，等高).\n\n又 $P, N, T, M$ 四点共圆, 故 $\\angle T N P+\\angle P M T=180^{\\circ}$, 由三角形面积公式\n\n$S_{\\triangle P M T}=\\frac{1}{2} P M \\cdot M T \\sin \\angle P M T$\n\n$=S_{\\triangle P N T}=\\frac{1}{2} P N \\cdot N T \\sin \\angle P N T$\n\n$=\\frac{1}{2} P N \\cdot N T \\sin \\angle P M T$\n\n于是 $P M \\cdot M T=P N \\cdot N T$.\n\n因为 $\\angle N C I_{1}=\\angle N C A+\\angle A C I_{1}=\\angle N Q C+\\angle Q C I_{1}=\\angle C I_{1} N$,\n\n所以 $N C=N I_{1}$, 同理 $M C=M I_{2}$. 由 $M P \\cdot M T=N P \\cdot N T$ 得 $\\frac{N T}{M P}=\\frac{M T}{N P}$.\n\n因 $N P=M C$, $P M=N C$.\n\n故$\\frac{N T}{N I_{1}}=\\frac{M T}{M I_{2}}$.\n\n又因$\\angle I_{1} N T=\\angle Q N T=\\angle Q M T=\\angle I_{2} M T ，$\n\n有\n\n$\\Delta I_{1} N T \\sim \\Delta I_{2} M T$.\n\n故 $\\angle N T I_{1}=\\angle M T I_{2}$, 从而\n\n$\\angle I_{1} Q I_{2}=\\angle N Q M=\\angle N T M=\\angle I_{1} T I_{2}$.\n\n因此 $Q, I_{1}, I_{2}, T$ 四点共圆.']			['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']	Multimodal	Competition	False				Theorem proof	Geometry	Math	Chinese
22	"如图, 在锐角 $\triangle A B C$ 中, $A B < A C, A D$ 是边 $B C$ 上的高, $P$ 是线段 $A D$ 内一点。过 $P$ 作 $P E \perp A C$, 垂足为 $E$, 作 $P F \perp A B$, 垂足为 $F。 O_{1} 、 O_{2}$ 分别是 $\triangle B D F 、 \triangle C D E$ 的外心。求证： $O_{1} 、 O_{2} 、 E 、 F$ 四点共圆的充要条件为 $P$ 是 $\triangle A B C$ 的垂心。

<image_1>"	['连结 $B P 、 C P 、 O_{1} O_{2} 、 E O_{2} 、 E F 、 F O_{1}$ 。因为 $P D \\perp B C, P F \\perp A B$, 故 $B 、 D 、 P 、 F$四点共圆, 且 $B P$ 为该圆的直径。又因为 $O_{1}$ 是 $\\triangle B D F$ 的外心, 故 $O_{1}$ 在 $B P$ 上且是 $B P$ 的中点。同理可证 $C 、 D 、 P 、 E$ 四点共圆, 且 $O_{2}$ 是的 $C P$ 中点。综合以上知 $O_{1} O_{2}$ $/ / B C$, 所以 $\\angle P O_{2} O_{1}=\\angle P C B$ 。因为 $A F \\cdot A B=A P \\cdot A D=A E \\cdot A C$, 所以 $B 、 C$ 、 $E 、 F$ 四点共圆。\n\n充分性：设 $P$ 是 $\\triangle A B C$ 的垂心, 由于 $P E \\perp A C, P F \\perp A B$, 所以 $B 、 O_{1} 、 P$ 、 $E$ 四点共线, $C 、 O_{2} 、 P 、 F$ 四点共线, $\\angle F O_{2} O_{1}=\\angle F C B=\\angle F E B=\\angle F E O_{1}$, 故 $O_{1} 、 O_{2} 、 E_{1} F$ 四点共圆。\n\n必要性: 设 $O_{1} 、 O_{2} 、 E 、 F$ 四点共圆, 故 $\\angle O_{1} O_{2} E+\\angle E F O_{1}=180^{\\circ}$ 。\n\n由于 $\\angle P O_{2} O_{1}=\\angle P C B=\\angle A C B-\\angle A C P$, 又因为 $O_{2}$ 是直角 $\\triangle C E P$ 的斜边中点,也就是 $\\triangle C E P$ 的外心, 所以 $\\angle P O_{2} E=2 \\angle A C P$ 。因为 $O_{1}$ 是直角 $\\triangle B F P$ 的斜边中点, 也就是 $\\triangle B F P$ 的外心, 从而 $\\angle P F O_{1}=90^{\\circ}-\\angle B F O_{1}=90^{\\circ}-\\angle A B P$ 。\n\n<img_4212>\n因为 $B 、 C 、 E 、 F$ 四点共圆, 所以 $\\angle A F E=\\angle A C B, \\angle P F E=90^{\\circ}-\\angle A C B$ 。于是, 由 $\\angle O_{1} O_{2} E+\\angle$ $E F O_{1}=180^{\\circ}$ 得\n\n$(\\angle A C B-\\angle A C P)+2 \\angle A C P+\\left(90^{\\circ}-\\angle A B P\\right)+\\left(90^{\\circ}-\\angle A C B\\right)=180^{\\circ}$ ， 即 $\\angle A B P=\\angle A C P$ 。 又因为 $A B < A C, A D \\perp B C$, 故 $B D<C D$ 。设 $B^{\\prime}$ 是点 $B$ 关于直线 $A D$ 的对称点, 则 $B^{\\prime}$ 在线段 $D C$ 上且 $B^{\\prime} D=B D$ 。连结 $A B^{\\prime} 、 P B^{\\prime}$ 。由对称性, 有 $\\angle A B^{\\prime} P=\\angle A B P$, 从而 $\\angle A B^{\\prime} P=\\angle A C P$, 所以 $A 、 P 、 B^{\\prime} 、 C$ 四点共圆。由此可知 $\\angle P B^{\\prime} B=\\angle C A P=90^{\\circ}-\\angle A C B$ 。因为 $\\angle P B C=\\angle P B^{\\prime} B$,故 $\\angle P B C+\\angle A C B=\\left(90^{\\circ}-\\angle A C B\\right)+\\angle A C B=90^{\\circ}$, 故直线 $B P$ 和 $A C$ 垂直。由题设 $P$ 在边 $B C$ 的高上, 所以 $P$ 是 $\\triangle A B C$ 的垂心.']			['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proof	Geometry	Math	Chinese
23	"<image_1>

如图, 锐角三角形 $A B C$ 的外心为 $O, K$ 是边 $B C$ 上一点 (不是边 $B C$ 的中点), $D$ 是线段 $A K$ 延长线上一点, 直线 $B D$ 与 $A C$ 交于点 $N$, 直线 $C D$ 与 $A B$ 交于点 $M$. 求证: 若 $O K \perp M N$,则 $A, B, D, C$ 四点共圆."	['用反证法. 若 $A, B, D, C$ 不四点共圆，设三角形 $A B C$ 的外接圆与 $A D$ 交于点 $E$, 连接 $B E$ 并延长交直线 $A N$ 于点 $Q$, 连接 $C E$ 并延长交直线 $A M$ 于点 $P$, 连接 $P Q$.\n\n因为 $P K^{2}=P$ 的幂 $($ 关于 $\\odot O)+K$ 的幂 $($ 关于 $\\odot O$ )\n\n$$\n=\\left(P O^{2}-r^{2}\\right)+\\left(K O^{2}-r^{2}\\right),\n$$\n\n同理\n\n$$\nQ K^{2}=\\left(Q O^{2}-r^{2}\\right)+\\left(K O^{2}-r^{2}\\right),\n$$\n\n所以\n\n$$\nP O^{2}-P K^{2}=Q O^{2}-Q K^{2},\n$$\n\n故\n\n$$\nO K \\perp P Q \\text {. }\n$$\n\n由题设, $O K \\perp M N$, 所以 $P Q / / M N$, 于是\n\n$$\n\\frac{A Q}{Q N}=\\frac{A P}{P M}. \\quad \\quad (1)\n$$\n\n由梅内劳斯（Menelaus）定理，得\n\n$$\n\\begin{aligned}\n& \\frac{N B}{B D} \\cdot \\frac{D E}{E A} \\cdot \\frac{A Q}{Q N}=1,  \\quad \\quad (2)\\\\\n& \\frac{M C}{C D} \\cdot \\frac{D E}{E A} \\cdot \\frac{A P}{P M}=1 . \\quad \\quad (3)\n\\end{aligned}\n$$\n\n由(1)，(2)，(3)可得\n\n$$\n\\frac{N B}{B D}=\\frac{M C}{C D}\n$$\n\n所以 $\\frac{N D}{B D}=\\frac{M D}{D C}$, 故 $\\triangle D M N \\sim \\triangle D C B$, 于是 $\\angle D M N=\\angle D C B$, 所以 $B C / / M N$, 故 $O K \\perp B C$,即 $K$ 为 $B C$ 的中点, 矛盾! 从而 $A, B, D, C$ 四点共圆.']			['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proof	Geometry	Math	Chinese
24	"<image_1>

如图, 给定凸四边形 $A B C D, \angle B+\angle D<180^{\circ}, P$ 是平面上的动点, 令 $f(P)=P A \cdot B C+P D \cdot C A+P C \cdot A B$.
求证：当 $f(P)$ 达到最小值时, $P 、 A 、 B 、 C$ 四点共圆;"	['如图, 由托勒密不等式, 对平面上的任意点 $P$, 有 $P A \\cdot B C+P C \\cdot A B \\geq P B \\cdot A C$. 因此\n\n$f(P)=P A \\cdot B C+P C \\cdot A B+P D \\cdot C A \\geq P B \\cdot C A+P D \\cdot C A$\n\n$=(P B+P D) \\cdot C A$.\n\n因为上面不等式当且仅当 $P 、 A 、 B 、 C$ 顺次共圆时取等号, 因此当且仅当 $P$ 在 $\\triangle A B C$ 的外接圆且在 $A C$ 上时, $f(P)=(P B+P D) \\cdot C A$. 又因 $P B+P D \\geq B D$, 此不等式当且仅当 $B, P, D$ 共线且 $P$ 在 $B D$ 上时取等号. 因比当且仅当 $P$ 为 $\\triangle A B C$ 的外接圆与 $B D$ 的交点时, $f(P)$ 取最小值 $f(P)_{\\min }=A C \\cdot B D$. 故当 $f(P)$ 达最小值时, $P 、 A 、 B 、 C$ 四点共圆.']			['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proof	Geometry	Math	Chinese
25	"如图, 在 $\triangle A B C$ 中, $P 、 Q 、 R$ 将其周长三等分, 且 $P 、 Q$ 在 $A B$ 边上, 求证: $\frac{S \Delta P Q R}{S \Delta A B C}>\frac{2}{9}$.

<image_1>"	['作 $\\triangle A B C$ 及 $\\triangle P Q R$ 的高 $C N 、 R H$. 设 $\\triangle A B C$ 的周长为 1 . 则 $P Q=\\frac{1}{3}$.\n\n则 $\\frac{S \\triangle P Q R}{S \\triangle A B C}=\\frac{P Q \\cdot R H}{A B \\cdot C N}=\\frac{P Q}{A B} \\cdot \\frac{A R}{A C}$, 但 $A B<\\frac{1}{2}$, 于是 $\\frac{P Q}{A B}>\\frac{2}{3}$,\n\n$A P \\leqslant A B-P Q<\\frac{1}{2}-\\frac{1}{3}=\\frac{1}{6}, \\therefore A R=\\frac{1}{3}-A P>\\frac{1}{6}, A C<\\frac{1}{2}$, 故 $\\frac{A R}{A C}>\\frac{1}{3}$, 从而 $\\frac{S \\triangle P Q R}{S \\triangle A B C}>\\frac{2}{9}$.']			['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proof	Geometry	Math	Chinese
26	"<image_1>


四边形 $A B C D$ 内接于圆 $O$, 对角线 $A C$ 与 $B D$ 相交于 $P$, 设三角形 $A B P 、 B C P 、 C D P$ 和 $D A P$的外接圆圆心分别是 $O_{1} 、 O_{2} 、 O_{3} 、 O_{4}$. 求证 $O P 、 O_{1} O_{3} 、 O_{2} O_{4}$ 三直线共点."	['$\\because O$ 为 $\\triangle A B C$ 的外心, $\\therefore O A=O B$.\n\n$\\because O_{1}$ 为 $\\triangle P A B$ 的外心, $\\therefore O_{1} A=O_{1} B$.\n\n$\\therefore O O_{1} \\perp A B$.\n\n作 $\\triangle P C D$ 的外接圆 $\\odot O_{3}$, 延长 $P O_{3}$ 与所作圆交于点 $E$, 并与 $A B$ 交于点 $F$, 连 $D E$, 则 $\\angle 1=\\angle 2=\\angle 3, \\angle E P D=\\angle B P F$,\n\n$\\therefore \\angle P F B=\\angle E D P=90^{\\circ}$.\n\n$\\therefore P O_{3} \\perp A B$, 即 $O O_{1} / / P O_{3}$.\n\n同理, $O O_{3} / / P O_{1}$. 即 $O O_{1} P O_{3}$ 是平行四边形.\n\n$\\therefore O_{1} O_{3}$ 与 $P O$ 互相平分, 即 $O_{1} O_{3}$ 过 $P O$ 的中点.\n\n同理, $\\mathrm{O}_{2} \\mathrm{O}_{4}$ 过 $P O$ 中点.\n\n$\\therefore O P 、 O_{1} O_{3} 、 O_{2} O_{4}$ 三直线共点.\n\n<img_4082>']			['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proof	Geometry	Math	Chinese
27	"<image_1>

已知 在 $\triangle A B C$ 中, $A B > A C, \angle A$ 的一个外角的平分线交 $\triangle A B C$ 的外接圆于点 $E$, 过 $E$作 $E F \perp A B$, 垂足为 $F$.

求证 $2 A F=A B-A C$."	['<img_4095>\n\n在 $F B$ 上取 $F G=A F$, 连 $E G 、 E C 、 E B$,于是 $\\triangle A E G$ 为等腰三角形, $\\therefore E G=E A$.\n\n又 $\\angle 3=180^{\\circ}-\\angle E G A=180^{\\circ}-\\angle E A G=180^{\\circ}-\\angle 5=\\angle 4$.\n\n$\\angle 1=\\angle 2$. 于是 $\\triangle E G B \\cong \\triangle E A C$. $\\therefore B G=A C$,故证.']			['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAeMBjQMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APfqKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooppagBcikJpvJpw96AExThSY9KUDFAC0UUUAFFFFABRRRQAUUUUAFFFI33TigBaK4vw94knv/ABtrGkzZCWgG3P1rtKACiiigApCcUE+lM5NADieKTrQBzS4oAAKd2pMUvagAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKTIoAXNJupueaO9ABk5oxQDilxmgAFLigCloAKKKKACiiigAooooAKKKKACiiigAooooA8tsP9A+KOrTdPtDhfrzXqVec67YGLxja3JbaJJh+PFeiE9KAFJppJo70GgA60uKAOKUDBoAMUtFFABRRRQAUUUUAFFFFABRRRQAVS1K6ntLQywQec4ydtXaRlDDacEe9AHFWXxG04ymHVHWzmzgITya660vre9hEsEgdD0NZeo+E9I1FWMllCJD/AB7eRXIXXgXVtKmN1pmq3Uvpb5wtAHpeaK81t/G2t6Q2zXtNW2t1/wCWxJOfzrrNH8X6NrgH2G7WQnsKAN6ijIxmigAooooAKKKKAEJApu/LYodc03pQA/NJ3oBpcUAJSgUuKWgBMUuKKKACiiigAooooAKKKKACiiigAooooAKM4opDzQA0vSg5phXmlU84FAHIeNIj/aWkSrwFnya62FxJEHHcVy3jmAta290rlfs7b8etbHhu6F/oNrcgcOuaANUClxS0UAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBVutPtb1dtzbxyr6OM1yer/Dy1vMtY3MlieuIFxXbUUAeXK3jDwqwit7U39svBlmbBA/KtrTfiRpU0gtr6XybvoUxnmu1dFkXawBB6g1ial4T0vUUI+zxxOf40UZoA17e6huYllicMrcjmpq8yufBGraFI1zo17cXT5yIpXIX6U618datpDBfEtnHbIP4ovm/pQB6XRWLpHinSdaj32lxkf7fFbKsrDKkEeo5oADTMc1Jx1pMUAAHFLQKKACiiigAooooAKKjnkEUDyHGFGTmvKrjxr46vNdvLPQ9HtJ7eA8O7kZH5Uk03ZA9FdnrNFeU/298V/+hdsf+/p/+JrtvC934gu7Zm16yhtph0EbZ/pVqLauS5WaR0FFFFSUFFFFABRRRQAUGiigBpoCilxSigDmvHUJbwnqEo/5ZxFqrfDSfz/AWls33vK5rY8Txef4Z1CL+9ERXMfDGXZpAsv+eCYx6dKAO9ooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACqtzp1pdj9/bRSe7KDVqigDhtX+HFpeSGe1u7m2cchIXKj+dY4l8Z+Gn2JBFPZKOXd8tivUaa6LIu11DD0NAHFaV8SNKvJRb3BkS5HDZjwAfrXYQ3cFwitFKjZ7BhmsnVvCmm6tGVeFYieN0ShT+Yrj5vA+reH3abw/cyTtnO25lJH5GgD02ivNYPHuo6K4j8SW+GzjMCZH6Cu00rxFp+rQiSGULkdHODQBrUUisrDKsCPY0tABRRRTS1DZXMXxZdG08M38yHDrCSteX+Dfib4f0myJvhci5dfnKwk5P1ruPiBcbNPitgcGfKYPernhfQLCHw/aJLYWzSBeS0QJrKhq5zXTQKsrKNJr+9+hj23xf8MXc6QxG73ucDMJAruLW5S7t0njzscZGeKgGj6ahBXT7UH1EQq4qhVCqAFHQAdK1VrW6ktS5/IWiiikUFFFFABRRRQAUUUUAFFFFAFbUIvPsJosZ3LivP/h5KU8Ua7Zn/ljgfrXpNeW+GGNh8S/EDdBcOAB+IoA9SooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAK1xY2typWWCN8jGSoNcXqnw0sppjd2c06XHYCTC/lXe0UAeX/AGvxl4Z4uRDJYJ2Qbmrb0b4jaZqTCJ4p4ZOhMqFR+ors3RJBhlDD0IzXP614T0jVkYXIEZ9UIWgDciureYZimjcf7LA1N+NeUz+HNS8Ond4e1K2SIHkXEwJ/nTrX4qJpMy22tnzZScboRlaANH4j+GfEOvy6fJojQhraTcRKetZkEHxaghWNP7L2r9K7zTvFWkajbrMl9AgYZw8oBFXRrGmMcDULU/SYUopRTS6jlJytfocNpg+KH9oxfbzp32Xd8+zGcV6HGH2Lv+9jnFV/7UsP+f22/wC/oqRL21kxsuYmz0w4qr6WJtrcnopqurDKsD9DTqQwooooAKKKKACiiigAooooAK8r1I/YPiDAw4NzOBn14NeqV5X8RkNr4y8LypwHujvP/ATQB6pRTUcOoYd6dQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRSEgKSTgVi6p4o03S0O+eOSRf4Fbk0AbdQz3dvbDM0yR/7xxXnNz4+1LV5DBpOnXdu3QSSJlfrTYPCHijWTu13UYpom6Kq7SB+dAHW6h4w0qxUlZ0nI7RuDXLXXxTSaU21lpN/5naQxkr/ACrY0/4ZeG7Ah0tpPM6kl8811Frp9raReXFCgUe1AHm51Hx1qPzWM8cKt0EqdKF8OfEO6x9v1OzYHrsXHH516iFUdFH5UtAHmqfDi5uf+P8AuN/+42Ka3wj02XiRSR67+a9MooA8q1H4KaPJZFbEzJcEdTMcZryy78JXfg/VWXUmnmiY4QRsa+qM1h+IdBtdZtw0iAzRcxt6GgDynQdI0nWk5jurbHeeQjP510S+CdRjUnSb+Lj7uW3f1rW03T7G9DaffRESxfLuzjcade6HrWlQM+j3Koi8bSMk0AYh0f4lWR3Q6pZeV3GzJP604eJPFekfNqSm4A6iGI5/nVmy+INzYyLaarpt079DKFwtdjZa/pGooPLuICx/gLAkUAcrp/xUs7pvLl0y9gYcFpEIB/Suqs/Eml3iArdwox/gZxmpb7Q9P1RMXEKspH8IxXK3vwt0IsbizheO67MX4FAHcxyxyrujdXHqpzT68ufSvGvh+TdBqEUlkvPlKhLfzq9p/wASdswtr/TLuNxwZXXCmgD0OiqFhrFjqKA21zHJ/sq2SKv0AFFFFABXnnxLgDXOlXP/ADwkLfoa9DrkPiJBu8M3Vz3gjLD2oA3tCn+06NbTHksuc1o1zngOc3PgrS5ScloQTXR0AFFFFABRRTCaAHE4pCabyacPegAWnUgpaACiiigAooooAKKKKACiiigAooooAKKOlVr2/trCEy3MyxqOctQBZrE1jxRp+kxkPMjzDpEG+Y1yup+L9R1i5NlocEignAu05Ue9XNI8BRmVbzW2W8u+u88YNAGTJrviTxTKYdKjm0xAcb5V4PvWtpfw7sxIt1q4Fzej/loD+ddtFGsMSxoMKowBT6AILe0gtYxHFGAo6DFTYxS0UAFFFFABRRSE0ALSEgU3JpTzQAmaMZpQMUYzQBz+vaYflvrYETRfNgdTVrQ9Zh1OAoJFM8YAlUHlT71r7cjBrzvW9Nm8K6v/AGxYAi3d/MuUX+P6/pQB3V7ptrqEJiuIgyEYPFcZf/Da1gcz6FttJzyWJzzXYaXqUOp2MVzER867iv8Ad9qvUAeXf2l4s8LN/wATAy6nGOghU8D866LSfH2n3wC3a/Yn/uzHBrriARg8iua1fwNoerlpLizVpT/FQBvQXdvdpuglSRT6c5qvqOjWOqRGK7hDqfwrg5/CviTQm83S9Vxar0t0HapbH4g31pN9m1TSblAOs74C0AOvvAV5pbmbwzcrZ4O4qcnNMsPG+paXcCy1ixuHI63G3C12Wn+JNK1JV+zXsLv/AHVbJFWr/TbbU7YwXUYkiI5B70AN0/VrLU491rcRy+ynOKvV5zf+D9Q0FzdeHbn7PCvJgUZLD0rQ0Pxykri21WF7OboDMcbqAO2rC8ZQ+f4P1SI87oSP1FbUciSoHRgynvVTWYftGj3UX99MUAcx8NZseGre0/54xgfSu1rzv4bTj+0NVtM/6hgtehk0ALSbqY77BuJ4FcrD41im1z+zfsrA5ID54oWrsOz3OsNJ1FAYFQacOlD3sIQClxS4ooAKKKKACiiigAooooAKKKKACiiigAoorlvEfihLHNlZAT3jHaVQ8rQ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proof	Geometry	Math	Chinese
28	"如图, $O 、 I$ 分别为 $\triangle A B C$ 的外心和内心, $A D$ 是 $B C$ 边上的高, $I$ 在线段 $O D$上。求证: $\triangle A B C$ 的外接圆半径等于 $B C$ 边上的旁切圆半径。

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注: $\triangle A B C$ 的 $B C$ 边上的旁切圆是与边 $A B 、 A C$ 的延长线以及边 $B C$ 都相切的圆。"	['由旁切圆半径公式, 有\n\n$r_{a}=\\frac{2 S}{b+c-a}=\\frac{a h_{a}}{b+c-a}$, 故只须证明\n\n$\\frac{R}{h_{a}}=\\frac{a}{b+c-a}$ 即可。连 $A I$ 并延长交 $\\odot O$ 于 $K$, 连 $O K$ 交 $B C$ 于 $M$, 则 $K$ 、 $M$ 分别为弧 $B C$ 及弦 $B C$ 的中点。且 $O K \\perp B C$ 。于是 $O K / / A D$, 又 $O K=R$,故\n\n$\\frac{R}{h_{a}}=\\frac{O K}{A D}=\\frac{I K}{I A}=\\frac{K B}{I A}$,\n\n故只须证 $\\frac{K B}{I A}=\\frac{a h_{a}}{b+c-a}=\\frac{B M}{\\frac{1}{2}(b+c-a)}$.\n\n<img_4201>\n\n\n\n作 $I N \\perp A B$, 交 $A B$ 于 $N$, 则 $A N=\\frac{1}{2}(b+c-a)$,\n\n而由 $\\triangle A I N \\sim \\triangle B K M$, 可证 $\\frac{K B}{I A}=\\frac{B M}{A N}$ 成立, 故证。']			['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proof	Geometry	Math	Chinese
29	"如图, 在 $\triangle A B C$ 中, $P$ 为边 $B C$ 上任意一点, $P E / / B A, P F / / C A$, 若 $S$ $\triangle A B C=1$, 证明: $S_{\triangle B P F 、} S_{\triangle P C E 、} 、 S_{\square P E A F}$ 中至少有一个不小于 $\frac{4}{9}\left(S_{X Y \cdots Z}\right.$ 表示多边形 $X Y \cdots Z$的面积).

<image_1>"	['<img_4144>\n\n如图, 三等分 $B C$ 于 $M 、 N$, 若点 $P$ 在 $B M$ 上 (含点 $M$ ), 则由于 $P E / / A B$, 则 $\\triangle C P E \\sim \\triangle C B A . C P: C B \\geqslant \\frac{2}{3}$. 于是 $S_{\\triangle P C E} \\geqslant \\frac{4}{9}$. 同理, 若 $P$ 在 $N C$ 上 (含点 $N$), 则 $S_{\\triangle B P F} \\geqslant \\frac{4}{9}$.\n\n若点 $P$ 在线段 $M N$ 上. 连 $E F$, 设 $\\frac{\\mathrm{BP}}{\\mathrm{BC}}=r\\left(\\frac{1}{3}<r<\\frac{2}{3}\\right)$, 则 $\\frac{\\mathrm{CP}}{\\mathrm{BC}}=1-r$.\n\n$$\nS_{\\triangle B P F}=r^{2}, S_{\\triangle P C E}=(1-r)^{2} . \\therefore S_{\\triangle B P F}+S_{\\triangle P C E}=r^{2}+(1-r)^{2}=2 r^{2}-2 r+1=2\\left(r-\\frac{1}{2}\\right)^{2}+\\frac{1}{2}<2\\left(\\frac{1}{3}-\\frac{1}{2}\\right)^{2}+\\frac{1}{2} =\\frac{5}{9}\n$$\n\n于是 $S_{\\square A E P F} \\geqslant \\frac{4}{9}$. 故命题成立.']			['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']	Multimodal	Competition	False				Theorem proof	Geometry	Math	Chinese
30	"如图, 设三角形的外接圆 $O$ 的半径为 $R$, 内心为 $I, \angle B=60^{\circ}, \angle$ $A<\angle C, \angle A$ 的外角平分线交圆 $O$ 于 $E$.

<image_1>
证明: $I O=A E$"	['$\\because \\angle B=60^{\\circ}, \\therefore \\angle A O C=\\angle A I C=120^{\\circ}$.\n\n$\\therefore A, O, I, C$ 四点共圆. 圆心为 弧$A C$ 的中点 $F$, 半径为 $R$\n\n$\\therefore O$ 为 $\\odot F$ 的弧 $A C$ 中点, 设 $O F$ 延长线交 $\\odot F$ 于 $H, A I$ 延长线交弧 $B C$ 于 $D$.\n\n<img_4222>\n\n由 $\\angle E A D=90^{\\circ}$ (内外角平分线) 知 $D E$ 为 $\\odot O$ 的直径. $\\angle O A D=\\angle O D A$\n\n但 $\\angle O A I=\\angle O H I$, 故 $\\angle O H I=\\angle A D E$, 于是 $R t \\triangle D A E \\cong R t$ $\\triangle H I O$\n\n$\\therefore A E=I O$']			['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']	Multimodal	Competition	False				Theorem proof	Geometry	Math	Chinese
31	"如图, 设三角形的外接圆 $O$ 的半径为 $R$, 内心为 $I, \angle B=60^{\circ}, \angle$ $A<\angle C, \angle A$ 的外角平分线交圆 $O$ 于 $E$.

<image_1>
证明: $2 R<I O+I A+I C<(1+\sqrt{3}) R$"	['$\\because \\angle B=60^{\\circ}, \\therefore \\angle A O C=\\angle A I C=120^{\\circ}$.\n\n$\\therefore A, O, I, C$ 四点共圆. 圆心为 弧$A C$ 的中点 $F$, 半径为 $R$\n\n$\\therefore O$ 为 $\\odot F$ 的弧 $A C$ 中点, 设 $O F$ 延长线交 $\\odot F$ 于 $H, A I$ 延长线交弧 $B C$ 于 $D$.\n\n<img_4222>\n\n由 $\\angle E A D=90^{\\circ}$ (内外角平分线) 知 $D E$ 为 $\\odot O$ 的直径. $\\angle O A D=\\angle O D A$\n\n但 $\\angle O A I=\\angle O H I$, 故 $\\angle O H I=\\angle A D E$, 于是 $R t \\triangle D A E \\cong R t$ $\\triangle H I O$\n\n$\\therefore A E=I O$\n\n由 $\\triangle A C H$ 为正三角形, 易证 $I C+I A=I H$.\n\n由 $O H=2 R$. $\\therefore I O+I A+I C=I O+I H>O H=2 R$.\n\n设 $\\angle O H I=\\alpha$, 则 $0<\\alpha<30^{\\circ}$.\n\n$\\therefore I O+I A+I C=I O+I H=2 R(\\sin \\alpha+\\cos \\alpha)=2 R \\sqrt{2} \\sin \\left(\\alpha+45^{\\circ}\\right)$\n\n又 $\\alpha+45^{\\circ}<75^{\\circ}$, 故 $I O+I A+I C<2 \\sqrt{2} R(\\sqrt{6}+\\sqrt{2}) / 4=R(1+\\sqrt{3})$']			['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']	Multimodal	Competition	False				Theorem proof	Geometry	Math	Chinese
32	"<image_1>

如图, 封闭多面体 $A B C D F E$, 平行四边形 $A B C D$ 中, $A C$ 与 $B D$ 相交于点 $O, E$ 在平面 $A B C D$ 的射影为 $O, E F / /$ 平面 $A B C D$, 且 $E F=\frac{1}{2} B C$, 其中 $A B=B C=A C=A E=a$.
求证: 平面 $A B C D \perp$ 平面 $C F D$;"	['取 $\\mathrm{CD}$ 的中点 $\\mathrm{M}$, 连接 $\\mathrm{FM} 、 \\mathrm{OM}, \\mathrm{OE}$\n\n$\\because E F / /$ 平面 $A B C D, E F \\subset$ 平面 $B C F E$, 平面 $A B C D \\cap$ 平面 $E F C B=B C$\n\n$\\therefore E F / / B C . \\because E F=\\frac{1}{2} B C$,\n\n$\\because \\mathrm{O} 、 \\mathrm{M}$ 分别为 $\\mathrm{BD}, \\mathrm{CD}$ 的中点, $\\therefore O M / / B C, O M=\\frac{1}{2} B C$\n\n$\\therefore O M / / E F, O M=E F \\therefore$ 四边形 $\\mathrm{OMFE}$ 为平行四边形, $\\therefore \\mathrm{OE} / / \\mathrm{FM}$\n\n$\\because O E \\perp$ 平面 $A B C D \\therefore F M \\perp$ 平面 $A B C D$\n\n$\\because F M \\subset$ 平面 $C F D \\therefore$ 平面 $A B C D \\perp$ 平面 $C F D$.']			['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']	Multimodal	Competition	False				Theorem proof	Solid Geometry	Math	Chinese
33	"<image_1>

如图, 在四棱锥 $P-A B C D$ 中, $P A \perp$ 平面 $A B C D$, 四边形 $A B C D$为矩形, $E$ 是 $P D$ 的中点, $M$ 是 $E C$ 的中点, 点 $Q$ 在线段 $P C$ 上且 $P Q=3 Q C$.
证明 $Q M / /$ 平面 $P A B$;"	['取 $P C$ 中点 $F, \\because E$ 是 $P D$ 的中点, $\\therefore E F / / C D$, 又由题意知 $Q$ 是 $F C$ 的中点, $M$ 是 $E C$ 的中点, $\\therefore E F / / Q M$,\n\n$\\therefore Q M / / C D / / A B$. 又 $Q M \\not \\subset$ 平面 $P A B, A B \\subset$ 平面 $P A B$,\n\n$\\therefore Q M / /$ 平面 $P A B$ .']			['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAVQCOQMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACimSSLEjO7BUUZZjwAK5d/iV4MjkaN/EdgrrwymTkUeQHV0Vm6Tr2l69A0+lX8F3EpwWibIBrSFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQA1yqqS2MAZOa8p8C2umX/izxLrU0Nt5Im8mPzAMLtLBjg8V3ni7WLbRPDd9d3MojAhZUyeSxHGPxxXF+A/A3h/VPB9neanYQ3V1OTO0m9gctzg4I96I6ty+QS2S+ZV8J+Tf/FnUr7QlK6MkBilZMiNpPl6Dp2NesE7QT261V0/TLLSbVbWwtkghUcKg4rzX45+K7nw94Yt7bT7ryrq7lKsF+9sweR+NO+ij2FbVy7nqiurruVgwPQg8U6vlf4dfF+98LSvZ6z595ZPwOfmjPc819G+H/Fui+JrVZ9LvopsjlAcMvtikM26KTNLQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAGVrfhvR/EcUUWsafDeRxEsiyg4BNS6RoWmaDafZdLs47WDOdkecZ/GtCkJPtRsBW1G/t9MsJr27kEcEK73Y9gK+Wr7UZfiJ43u9Z1ORk0PTjvkbqEiDYAH1J/Wu0+OPjWW6nj8IaSTJJKR9o8vknPRfxz+lXNR8HReDPgNq0JQfbLmBXuH9964/QCgDgviJ4a0nUrI+LvCaiTSzIYp0RTiNhjnHvkflWX4e8Oa5Lpa694TvpGng4mt43+eP354x1r2D4Hadbaj8K7izu4xJDPdSB1YZHKrXns8Wo/Bj4i+dGXbS5zxjoyE9PqB/OgDpfCnx1ubGaPTfFtpIjj5ftGCMe7A/0Fe2aTrum65ZpdabeRXMTjIKN/TrXKax4L8K/EbSItQMKeZOgZLmLAcHHTPtXkOq/D/wAafDa9fUvD91LPag7m8k849GHU0AfTQ5orxLwf8eLeYpY+KLdrS56GcKQv4jrXsdjqNpqNstxaXEU0TDIZGyKALVFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFACE1yvxA8XQeD/C1zfuw89l2QL3LHpXTzSpDE8sjBURSWJ7Ac18zeJ9RvPiz8TItIsGc6bBJsHoAD8zfjjigDW+DPhKfxDr8/jLV1MgWQtBv7uTww9hzXpXxhZV+GGrj1iAH/fS11ui6RbaHpFtp1ogSKFAoAGPqa4340D/AItrqX+6P5igDO+ASkfDnnvdOR+S11Hj7wba+MvDk1lIi/aVUtBIeqtjj8K5z4EcfDiL/ru/8lr00jNAHzl8J/GN14P8RS+EdcLRwtKUjMnSNs/yPWvowbZEznKmvHfjX8PDqtkfEelxkX9suZQg5cevHcY/Wr3wc+IQ8SaT/ZGoSAalaDbljgyL60Aa3jD4SeHvFKSTLbraXx5WaIBQT6sAOa8eutB8f/Ci7M+nzPcWAPWMFkI91zx9a+oByM02SJJkZJFDowwVYZBoA8m8GfHPSNa8uz1dDY3vClm+659favV4LiK5iEsMqSxsMhkbINeZ+NPgpoXiEPc6ePsF6ckFP9WT/u15etx8QfhJehJBJc2GcYwZIyv8l6UAfUIOaK828GfGTQfEwS3uXFjfEcxyn5T9G6V6Orq6hlYFT0I5zQA6igUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFITS1j+JtetfDWg3Wp3TqqwoSoJ6ntQB5v8bvHR0jS10HTpf9Pu+H2nlFP+PIq/8GfAo8NeHV1G7j/4mF6ockjlUIyB7dea84+Heg3nxJ8e3HiXWEZrSJ953cgt2UfTrX0uqhVAUYA6AUAKOKztb0Ox8Q6ZJp+oxmS3kGGUHFaNFAGXoPh/T/DenCw0yIxW4YttJzyf/wBValFFADJEWRSrKGBGCD6V82fEjwtefDrxfb+KdF3LaSShyFzhD3B9sD9a+lqzde0W08QaNc6beRh4pkK8jOD2NAGd4L8WWvi/w9BqNuwDsMSx55Ru4roxXzBoOoah8HPiJJpl8XbTLhgpY/dZD0YfjX0xaXUV5aRXMDiSKVQysD1BoAnqC6tLe9gaG5hSaJhgo6hgfwNTjmigDxvxl8CNN1Mve6A5srvJYxk/Ix9v7tcHp/i7x18LbsWerQS3NipxiQllI/2XNfT+Kqajpljqts1vfWsdxGw+66g/lQByfg/4peHvFsSJFcLb3h+9BKdvPoCetduDmvCvF3wGUM9/4UuWhuQdwgdsD8GzkfgK53Rvid4v+H10ml+JbSS4tVOwecCGx6hiMtQB9MUVy3hbx/oHi2FDp96nnsMmCQ7XH4V1GaAFooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAEJx+Wa+bvir4luPHXjK08KaOxeCKXYdh4du5/Afyr0z4u+OE8KeGXgtpduo3ilIgOqjBy34cVzfwO8DvaWsnibU4ibq7z5O8cqM8n65BoA9L8H+Gbbwp4bttLtwMxr+8cD77Hqa36BRQAVzvjHxdaeDdIGo3kcjxltuEXJroq8o+P2P+EFj95v6UAd/wCGPENv4o0KDVbVGWGYZUMMGtiuC+Dv/JNdM/3P6Cu9oAKMUUUAcF8UvAkPjLw8/lRqNQtgXhfHPuv44rz/AODPjyawvH8H64xjlRituXPRh1X+WK98Irwj40eApbeceLdDQpNEwadYhgg5yG/nQB7uKWvPfhX4+j8YeH1imcLqNsAkiE5LDs38s/WvQhQAUUUUAGKytb8O6V4gtGttSs450YYyRhh9D1rVoIzQB89+KfgdqWkTtqPhC8l+U7hBuKsPYHv+NVvDfxn1zwvejSfFtk7pH8rMRiRR9O/1r6MwKwPEfgzRPFNq0GpWSOT0kAwwP1FAD/Dvi/RfFFqJ9LvY5eMtHuG5fqK3Aa+b/EHwg8S+ELttT8J3k08cZ3CNDiRfw6EVpeE/jrdWE403xbassiHY04B3Z/2h0oA9/orN0fXtM16zW6028iuYW7o2a0hQAUUUUAITiudvPHnhXTrt7W81+wgnjOGjeYAiuiNeP67qvhuL4wLJq11aW8FlbHd5ygB3cZHbk8Un8SQ18NzvLf4heELu4jt7fxFp8s0jbURJgSxrpAQwBHIPSuL0y00PxraxajDZWy20FyJLaaKMLv25GeK7RQFUAdKt7E9RaKKKkYUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFU9U1G30nTp766cJDAhdiTjpVsnFeBfG7xnPqV9D4P0gmR3YeeEP3m7L+fWgDndMtrz4wfFCS6ulY6XbsdwP3QgIG36nr+FfTVtbx2tvHBCoWOMBVA6Yr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proof	Solid Geometry	Math	Chinese
34	"<image_1>

如图, $E A \perp$ 平面 $A B C, A E / / C D, A B=A C=C D=2 A E=4$, $B C=2 \sqrt{3}, M$ 为 $B D$ 的中点.
求证: 平面 $A E M \perp$ 平面 $B C D$"	['取 $B C$ 中点 $N$, 连接 $M N, A N, \\therefore M N=A E, M N / / A E$,\n\n<img_4099>\n\n$\\because E A \\perp$ 平面 $A B C, \\therefore$ 四边形 $A E M N$ 是矩形,\n\n$\\therefore E M \\perp M N,$\n\n由题意知, $E D=E B=2 \\sqrt{5}, \\because M$ 为 $B D$ 的中点,\n\n$\\therefore E M \\perp B D$,\n\n又 $\\because B D \\cap M N=M, \\therefore E M \\perp$ 平面 $B C D$,\n\n$\\because E M \\subset$ 平面 $A E M, \\therefore$ 平面 $A E M \\perp$ 平面 $B C D$.']			['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']	Multimodal	Competition	False				Theorem proof	Solid Geometry	Math	Chinese
35	"<image_1>

如图, 在四棱锥 $A-B C D E$ 中, 平面 $A B C \perp$ 平面 $B C D E ; \angle C D E=\angle B E D=90^{\circ}, A B=C D=2$, $D E=B E=1, \quad A C=\sqrt{2}$.
证明: 平面 $A C E \perp$ 平面 $B C D E$;"	['$\\because C D=2, D E=B E=1, \\angle C D E=\\angle B E D=90^{\\circ}$,\n\n$\\therefore B C=\\sqrt{2}$,\n\n$\\because A B=2, A C=\\sqrt{2}, \\therefore A C \\perp B C$\n\n$\\because$ 平面 $A B C \\perp$ 平面 $B C D E$, 平面 $A B C \\cap$ 平面 $B C D E=B C, A C \\subset$ 平面 $\\mathrm{ABC}$ 且 $A C \\perp B C$\n\n$\\therefore AC \\perp$ 面$BCDE$\n\n$\\because A C \\subset$ 面 $A C E \\therefore$ 平面 $A C E \\perp$ 平面 $B C D E$.']			['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']	Multimodal	Competition	False				Theorem proof	Solid Geometry	Math	Chinese
36	"如图, 多面体 $A-P C B E$ 中, 四边形 $P C B E$ 是直角梯形, 且 $P C \perp B C, P E \| B C$, 平面 $P C B E \perp$平面 $A B C, A C \perp B E, M$ 是 $A E$ 的中点, $N$ 是 $P A$ 上的点.

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若 $M N \|$ 平面 $A B C$, 求证: $N$ 是 $P A$ 中点;"	['记 $P A E \\cap A B C=l$, 因为 $M N \\| A B C$, 根据线面平行的性质定理, 有 $M N \\| l$.\n\n又 $P E \\| B C$, 所以 $P E \\| A B C$, 从而 $P E \\| l$, 于是 $P E \\| M N$.\n\n又 $M$ 是 $A E$ 的中点, 于是 $N$ 是 $P A$ 的中点.']			['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proof	Solid Geometry	Math	Chinese
37	"<image_1>

如图, 在三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, 侧棱 $A A_{1} \perp$ 底面 $A B C$, $A B=A C=2 A A_{1}, \angle B A C=120^{\circ}, D 、 D_{1}$ 分别是线段 $B C 、 B_{1} C_{1}$ 的中点, 过线段 $A D$ 的中点 $F$ 作 $B C$ 的平行线, 分别交 $A B 、 A C$ 于点 $M 、 N$.
证明: $M N \perp$ 平面 $A D D_{1} A_{1}$;"	['因为 $A B=A C, D$ 是 $B C$ 的中点, 所以 $B C \\perp A D$,\n\n因为 $M N / / B C$, 所以 $M N \\perp A D$,\n\n因为 $A A_{1} \\perp$ 平面 $A B C, M N \\subset$ 平面 $A B C$,\n\n所以 $A A_{1} \\perp M N$,\n\n因为 $A D, A A_{1} \\subset$ 平面 $A D D_{1} A_{1}$, 且 $A D$ I $A A_{1}=A$,\n\n所以 $M N \\perp$ 平面 $A D D_{1} A_{1}$.']			['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proof	Solid Geometry	Math	Chinese
38	"<image_1>

如图, 正方形 $A D E F$ 与 $\square A B C D$ 所在的平面互相垂直, 且 $A B=2 A D=2 a$, $\angle B A D=60^{\circ}, G$ 为 $B D$ 的中点.
求证: 平面 $B D E \perp$ 平面 $B C E$ 。"	['$\\because A B=2 A D=2 a, \\angle B A D=60^{\\circ}$,\n\n$\\therefore$ 在 $\\triangle A B D$ 中, $B D^{2}=4 a^{2}+a^{2}-2 \\cdot 2 a \\cdot a \\cdot \\cos 60^{\\circ}=3 a^{2}, B D^{2}+A D^{2}=A B^{2}$,\n\n$\\therefore \\angle A D B=90^{\\circ}, B D \\perp A D$,\n\n又 $A D E F$ 为正方形, $\\therefore A D \\perp D E$,\n\n又 $A D / / B C, \\therefore B C \\perp D E, B C \\perp B D$,\n\n又 $B D \\subset$ 面 $B D E, D E \\subset$ 面 $B D E, B D \\cap D E=D$,\n\n$\\therefore B C \\perp$ 平面 $B D E$,\n\n又 $B C \\subset$ 平面 $B C E, \\therefore$ 平面 $B D E \\perp$ 平面 $B C E$.']			['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']	Multimodal	Competition	False				Theorem proof	Solid Geometry	Math	Chinese
39	"<image_1>

如图, 正三棱柱 $\mathrm{ABC}-\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$ 的所有棱长都为2 , 点 $M, N$ 分别是 $B A^{\prime}$ 与 $B^{\prime} C^{\prime}$ 的中点.
求证: $M N / /$ 平面 $A C C^{\prime} A^{\prime}$."	['连接 $A C^{\\prime}$,\n\n<img_4094>\n\n$\\because M, N$ 分别是 $B^{\\prime} A$ 与 $B^{\\prime} C$ 的中点\n\n$\\therefore M N / / A C^{\\prime}$, 又 $\\because M N \\not \\subset$ 平面 $A C C^{\\prime} A^{\\prime}, A C^{\\prime} \\subset$ 平面 $A C C^{\\prime} A^{\\prime}$. 所以 $M N / /$ 平面 $A C C^{\\prime} A^{\\prime}$;']			['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FFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFZet6Dp/iCzNrqEAkTqD0Kn1BrUooA4q0+GWiWk6u1zf3CKcrFPMpQfhtH867KNFjRY0GFUAAegp9FABRRRQB//Z']	Multimodal	Competition	False				Theorem proof	Solid Geometry	Math	Chinese
40	"如图，已知长方形 $A B C D$ 中, $A B=4, A D=2, M$ 为 $D C$的中点, 将 $\triangle A D M$ 沿 $A M$ 折起, 使得平面 $A D M \perp$ 平面 $A B C M$.

<image_1>
求证: $A D \perp B M$."	['连接 $B M \\because A B=4, A M=B M=2 \\sqrt{2}$\n\n$\\therefore B M \\perp A M$\n\n$\\because$ 平面 $A D M \\perp$ 平面 $A B C M$,\n\n平面 $A D M \\cap$ 平面 $A B C M=A M, B M \\subset$ 平面 $A B C M$ ，\n\n$\\therefore B M \\perp$ 平面 $A D M$\n\n又 $\\because A D \\subset$ 平面 $A D M \\therefore A D \\perp B M$.']			['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAR0EAAMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoopMigBaKMijNABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUZoAKKKM0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUZ7d6M0AFFFGaACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooozigAoozSZoAWiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigBDXnPxAvvGWkk3ekXdulr0CuuT/ADr0euO+I3/IvH6n+VAHF6HN8U9d0qO/t9TsFjc4wyc/zqGHx/4p8K681n4qxcQEhVeBMLn613nw1GfBVr6bmpfiHo9tqXha7eZAXijLKcd6AN/R9Xtta06K9tWzHIMgZ6Vfrxb4Oa+troVxFdShYrWINyfpXrml6nDqtlHdQcxuMg5zmgC7RRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFACGvIPiFP4z0W5a5steEduxLCML0FewV5r8Wv+QfH/ALhoAztO8PfEe/02C8XxciiVN20oeKy9M8feIvCmrNYeKBNcoWIWdhhcfia9Y8M/8i1p/wD1yFcX8ZdLguPB9xdsg86PAVvzoA9Csb2HULOK5hYMjqDxVkHNeffCO/kvvCp8wk+WwUZ9Oa9BFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBk+IbbUbrTHTTLsWs4BIkrxnSPHXiXQvGf2HXtSa5tD8oYjjJOBXvE3+pkP8AsmvDvFXh/wDtKze4hi3TwzFyR6Kc/wBKAPcYJRNBHIpyGUEYrjviP4ln8PaEZrOTbcEkcdRxR8O/EI1fw0HkOJISykH/AGa5nxkjaxcXzt81skTbfTIzQAvwyuPF2v2q6tqGtiW1kJAhK9MV6yOlec/BkY8EwD0dv6V6PQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAVHPMlvC8shwijJNSVyHxG1CXT/C87REgujLx+FAHKa14x1nxLrbaR4YkktjC3z3K8qRU8lt468Mxtd3urPqqDnyolINL8FrNR4aivWj/fSp8znvXqDAMpU9DxQBxfgTxLeeIZLx7pHiKAEQv1Wu2qlaaVZ2NxLPbwKkkv3yO9XaACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACuO+I3/IvH6n+VdjXBeM/A+teJ5mFtrxtbY4/dFf/AK1AF74af8iVa8fxN/Ss74l+LLHTNEnsFnRrqZSnlg8isWw+GXi/TLRbW08ZtHCvIUIf8K5nxD4OfR73fq+pDV9Quj5cWRzE3Y0AYHgrT7/URDY225Ff5btR/d969Ck1XUvA7TR2yyXenW3/ACzjH3F966nwB4TTQ9IWWdB9tlXErkcmuZ8TQaj4e1qWaYvc6ZftiZeixqPWgDe0H4o6Tq8atcFbMEf8tHFdnaahaXybrWdJVxnKHNeIz+AIZ7q31HT1F7ZTHcI0HCD3rpP+EK16wt47vS9Ve2TG42wXn6UAepZpa8utfHGvaPKLbU9FuXQcG4YjH1611+meNdE1JVVL6ITHgx55BoA6Kimq4dQynIPQ06gAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigArzX4s82MY/2DXpJ6V4H40/4RfU9auY77xtcQPHIQ0IAwntQB7N4ZI/4RjTyenkg15j8VvFdvqls3h7TZFnuJTgqvUEHFcnDL4dgiWKP4l3qovAUAYA9K3PCuq/Djw88k9zrseoXTNkTTKNwoA9J+H+gNoPhuOJvvygOwPUV1lcGvxf8AA6qFGsRgDoAKX/hcPgj/AKDMf5UAd3RXCf8AC4fBH/QZj/Kj/hcPgj/oMx/lQB3dFcJ/wuHwR/0GY/yo/wCFw+CP+gzH+VAHd0Vwn/C4fBH/AEGY/wAqP+Fw+CP+gzH+VAHd0Vwn/C4fBH/QZj/Kj/hcPgj/AKDMf5UAd3RXCf8AC4fBH/QZj/Kj/hcPgj/oMx/lQB3dFcJ/wuHwR/0GY/yo/wCFw+CP+gzH+VAHd0Vwn/C4fBH/AEGY/wAqP+Fw+CP+gxH+VAHbz8wyD/ZP8q4rw/ClzLdxOoKv5gwffP8AjWR4g+JXg7WdNa1g8S/ZXJ+/GBmuI0DUfC2k6h9rbx5cSDdny2AwRQBbi1WbwL4xuNMkUrazoQnYbm//AFiu1u7E2vgCZ5lPmsrksx5AIyKw9c8VfDnXbmG4udZi82Mqd23JOCP8Kq+LPE/g3xLbLbxeL3s4hgFYwMEelAHT/BrjwVB/vv8A0r0YV4B4R1Pwf4UuN0XjeaaEdIWAC16IPjB4IA/5DMdAHd0Vwn/C4fBH/QZj/Kj/AIXD4I/6DMf5UAd3RXCf8Lh8Ef8AQZj/ACo/4XD4I/6DMf5UAd3RXCf8Lh8Ef9BmP8qP+Fw+CP8AoMx/lQB3dFcJ/wALh8Ef9BmP8qP+Fw+CP+gzH+VAHd0Vwn/C4fBH/QZj/Kj/AIXD4I/6DMf5UAd3RXCf8Lh8Ef8AQZj/ACo/4XD4I/6DMf5UAd3XO+NNLOq+HLqJATIsZKgVjf8AC4fBH/QZj/Kkb4v+B2BB1iIjHIIoA5f4SeIE06M+HNQIgnt1wA3XNeu3d3DZWslzcSBIkGWY9hXlk+meFviNftJ4d1f7New/PJLAo3H61es/hbqEN3HJdeKr67hU/PDLjDj0PFAHY6Hrp1uSd44QLUf6qYHhxW3VezsoLC2S3t41SNBgKO1WKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiio5pVhieRzhFGSfagChrurw6PpstzK4DBTsU/wAR9K4HwfpE/iTWJPEWoqxhkw0MT8hCDVfUri48deKv7PtmK2Nqwk3r0f1B/SvUbKzhsLVLe3jEcSDhRQBOBxVPVNMt9WsZbW4QMki7TmrtFAHk1hd3ngHXzZXZZ9NuGxEW4WJa7rXRdaloLT6TcurumUKfxVN4i0C217TJLeZQJCuEkxyv0rhfDGuXfhfWToWrlhE77LV3P3wO9AFzQfFH224XQ/EVkI7j7ieYeZPetLVPh1pd3mTTkjsZj/y0jUZzXStpWnXN1FfNaRPOnKSEcis3xF4lTw4FkmjAthjc/pQBxbaT4u8LOZILq51ZAciM8CtOw+JaxOI9etF04jg72rstI1iz1qzW4tJRIhAzjtUd/wCHdK1NSLqxikJ7sKAH6brum6ugayulmB/u1pZrznUvhtOshn0rVp7UjpHGOKpRa14v8Nt5dzpZuLVes7tzQB6nRXGaV8SdCvnW3muljuj1jrrobiKeMSRsCh6GgCWiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigBDXjvw50bTtU8W+LWvrKC4KXmFMiZxwK9iNeV/Cr/AJGvxh/1+f0FAHdDwj4f/wCgPaf9+xS/8Ij4f/6A9n/37FbQ6UUAYv8AwiPh/wD6A9n/AN+xR/wiPh//AKA9n/37FbVFAGL/AMIj4f8A+gPZ/wDfsUf8Ij4f/wCgPZ/9+xW1RQBi/wDCI+H/APoD2f8A37FH/CI+H/8AoD2f/fsVtUUAYv8Awi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proof	Solid Geometry	Math	Chinese
41	"<image_1>

如图, 平面四边形 $A B C D$ 中, $\triangle A B D$ 为等边三角形, $B D=2$, $B C=C D=\sqrt{2}$, 沿直线 $B D$ 将 $\triangle A B D$ 折成 $A^{\prime} B D$.
当 $A^{\prime} C=2$ 时, 求证: 平面 $A^{\prime} B D \perp$ 平面 $B C D$;"	['取 $B D$ 中点 $M$, 连接 $A^{\\prime} M, M C$.\n\n<img_4117>\n\n$\\because \\triangle A B D$ 为等边三角形, $B D=2, \\therefore A^{\\prime} M \\perp D B$,\n\n$A^{\\prime} M=\\sqrt{3}$,\n\n$\\because B C=C D=\\sqrt{2}, \\therefore M C=1$,\n\n当 $A^{\\prime} C=2$ 时, $A^{\\prime} M^{2}+M C^{2}=A^{\\prime} C^{2}$,\n\n$\\therefore A^{\\prime} M \\perp M C$,\n\n又 $D B \\cap M C=M, D B, M C \\subset$ 面 $B C D$\n\n$\\therefore A^{\\prime} M \\perp$ 平面 $B C D$ ，\n\n$\\because A^{\\prime} M \\subset$ 平面 $A^{\\prime} B D ， \\therefore$ 平面 $A^{\\prime} B D \\perp$ 平面 $B C D$.']			['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']	Multimodal	Competition	False				Theorem proof	Solid Geometry	Math	Chinese
42	"<image_1>

如图, 在四棱锥 $\mathrm{P}-\mathrm{ABCD}$ 中, $\mathrm{PD} \perp$ 平面 $\mathrm{ABCD}$, 底面 $\mathrm{ABCD}$ 是菱形, $\angle \mathrm{BAD}=60^{\circ}$ $A B=2 a, P D=2\left(1-a^{2}\right)$, 其中 $0<a<1,O$ 为 $A C$ 与 $B D$ 的交点, $E$ 为棱 $P B$上一点.
求证: 平面 $\mathrm{EAC} \perp$ 平面 $\mathrm{PBD}$;"	['$\\because P D \\perp$ 平面 $A B C D, A C \\subset $ 平面 $A B C D, \\therefore A C \\perp P D$\n\n$\\because$ 四边形 $A B C D$ 是菱形, $\\therefore A C \\perp B D$\n\n又 $\\because P D \\cap B D=D, \\therefore A C \\perp$ 平面 $P B D$\n\n而 $A C \\subset $ 平面 $E A C, \\therefore$ 平面 $E A C \\perp$ 平面 $P B D$.']			['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']	Multimodal	Competition	False				Theorem proof	Solid Geometry	Math	Chinese
43	"<image_1>

如图, 在斜三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, $A C=B C=A A_{1}=a, \angle A C B=90^{\circ}$,又点 $B_{1}$ 在底面 $A B C$ 上的射影 $D$ 落在 $B C$ 上, 且 $B C=3 B D$.
求证: $A C \perp$ 平面 $B B_{1} C_{1} C$;"	['$\\because$ 点 $B_{1}$ 在底面 $A B C$ 上的射影 $D$ 落在 $B C$ 上,\n\n$\\therefore B_{1} D \\perp$ 平面 $A B C, \\because A C \\subset$ 平面 $A B C, \\therefore B_{1} D \\perp A C$,\n\n又 $\\angle A C B=90^{\\circ}, \\therefore B C \\perp A C$,\n\n又 $B_{1} D \\cap B C=D, B_{1} D, B C \\subset$ 平面 $B B_{1} C_{1} C \\therefore A C \\perp$ 平面 $B B_{1} C_{1} C$.']			['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iycbjqGoPBbMeT5EZIGPY8D/gNem1zngbSl0bwXo9kAdyWqNJnqXYbm/UmujoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigApCeeOtLXN+O9f/wCEZ8GarqqsBLDARDn/AJ6MdqfXkg/hQBh/D2yF3rfinxSwDf2lqDwW0nXMMJ8sEexIP5Cu/HSuZ+HmmtpXw/0O0kVllFoskgbqHf52z+LGunoAKKKKAENcF8Jhjw9qq/3davB/5ErvTXCfCoY0bWx6a7e/+h0Ad5RRRQAUUUUAFFGaTcvHI56UALnHWmk/TpWN4i8TaT4ashd6neRQIeFBOXk9kUZLH2A/EVxVpL4x+IUkruX8P+FZshMLtvbmP1ychAfXA49etAG1rXxBtLbVG0XRLOfW9aAx9ntMeXGf+mkhOEH5njtkVk6R8PdT1bU11rx1qLajMG3RaXG3+hwegK9H/wA53V1vhvwjo3hOwNno9gtvG5zIclmkPqzHk1vCgCK3gjt4VihjWONRhUVQoH0A4H4VN+FFFAHM+OPDt34k0E22n6lcadfROJraeGQoPMHQPjqpz+GAeehp+DfGsOvmXSr1WtNfsAEvrOQFSrg4LIejIT09iK7Bhk/hisv+wdNXxB/bgs411LyPINwOCUyDg9uo60Aao6c8UE4rK0bXtO1yK4bT5xJ9mneCZSu0o6nByDyB6HvWoTQB5z8V777Rb6L4WgLG51u/ijZVP/LFGBcn8x+Ga9FjxtwOAOw7V51DY/278b7u/mzJBoFgkMI/h8+UEn8QjfqK9GTO0butADq5zx3ov/CReCtX0tUDSzWzeUD/AM9Byn/jwFdHTWHXj86AOY+HV+dR+HmgXTsWY2aRsSeSyDa2fxU11NcH8N7hLca/4fACjSNVmjhTPIhkJkT+bflXdjpQAtFFFABXlvxYkbWtR8N+DLc5k1O9E1wuekEfLZ+vP/fNeomvPNNshq/xp1jV5Bui0iyhsYD2EjjzGx7gNg/7woA9BRFRQqjCgYAA6U+kFLQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABXnHxPsm1/U/CfhoH93e6kZ58f88oULMD/AN9fnivRWOPr2rz3Rb5te+MWuSg77XRLOOyiI5USSNucj3+UA/TFAHoKjAwOg6Yp9IvSloAKKKKAE71wvwv403xAvpr97/6GK7uuG+GYxaeJR6eILz+a0AdzRRRQAUE00tjntXI+KfiBpnh6T7Fbq+o61IMQafagvIWPTdj7o+tAHVSyrGjO7KqKMlmOAPqewrzuX4gXnia/utH8E2DXTISkmsXBxaQH1HUyfT6dRS6P4Q17xLE9x8QL0XEEvzR6PbOY4Yv98qQXPHQkj613un6bZ6VYxWVjbx29tCMRxRrhVHsKAOQ8K/DuHR9Rk1rV9Qm1nXZet3coMRD0jX+Hnv8AoK7iNNoPqTmnAYpaACiiigAooooAKYy7jjsRg/Sn0UAed+K/DOr2XiVPGnhqTdexxhL6wJwL2JecD/bA4Hrgc9c9vc3kdjpkt9c5ihhiMsnQ7VAyRx7CrZUd68k+O2patY+HLKHT7wx21/K1pcwJGGeVSNw2nGQMKwOPUUAb/wAJZJtQ8KXWu3CFZtZ1Ce9IP8ILbFH0ATiu9HSsTwlPpM/hfTjoc0cunLAiwGPsoGMEdj6jqDnNblABSGlooA8ugZvD37QNxC3/AB7+IrBXTB4EsQOf0Rvzr1AGuC+IdotvqfhbxCoCyafqscUkn92GY7Gz+JX8zXeJ0/p6UAOooooArX95Dp9jcXtw22G3iaWRvRVBJ/QGuI+EbzXng+XWrhSsusahcXrA9QC+0D6AIPwp/wAXprr/AIQSbT7JmF3qdxFYxBerF3GV/FQR9K67R9Lt9G0ez021QLBaxLEgHoBigC+KKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAztc1SLRNEv9UmBaO0t3nZRxuCqTj9P1rjPg3pc1p4J/tS7+a81m4e/lYggncflzn2Gf8AgVX/AIoSed4ROjRS7LnWLmGwiA6ne43fgEDGuusrWGysoLW3jEcEMaxxoOiqBgD9KAJhwKWiigAooooAK4j4cDCeKV9PEV3/AOy12+a4j4fsqHxbuIAHiK6PP+6hoA7Y+x5rK13XtN8PabJqGqXcVvbxjlnYZY88KO5PoK5HWPiDLqgl0zwJbf2zqZO1rgD/AEW1HdnckBj6KD/gZdG+HizPDqXjKY69rS4YSSk+Tb9Pljj4GOOTjJ4oAzZbzxX8SLBV0qOfwvo8nLXtwu65uUPZEBGwY755yMd667wl4O0vwjpX2PT4SXc75riUhpZmPJLt3+nT+Z6FBgdD+NOoAaqnqepp1FFABRRRQAUUUUAFFFFABRRRQAh45rgL2WHW/i7p1kMFdAsZLuQ9QZZtqKMeoUE/8CrvJZI443eRgERSzE9AK81+EUU+q/2/4wu1fzdZvm8kv18iPhAPbkj/AIDQBs6J4XvfD/i2/udMureLQb5TNLYYJMVxwC0fYAgc+9dmOM5NMZcAkJz14HU/5Nee3uual4J8ZSzeIdRkl8L6oQttcOgC2Mwydj45CkZwTnoM9zQB6PRUVvKksCSI6sjgFWVshh2IPcYqWgDkviZYtf8Aw316Jc71tWmXHXKfP/7LWj4Q1pPEPhPTNWXrcW6s49HHDD8CCK17iGOeGSKZQ0ciFHU91PBFedfB1ZtO0jWvDlwW36PqksEasekTYZf13H8aAPShRSDpTX6/560AcHr14uqfFrw1oSgOlhDNqk46jO0xx/kST+Vd8vSvMPh5FJrfjXxd4sm3Or3Z0+yc9oouuPY/L+NenigBaKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAprHHv3p1MkAI56UAeZ65cP4g+OOhaQhzbaHbSX8//AF0cbVB+nykf7xr05fuivP8A4e2Qu9a8VeKGAb+0tReGB+pMMP7sEfUg/wDfIr0Begz1oAWiikNAC0hYAgZ5PSqWp6pZaRp8+oahOlvaQLvklkPCj/HmvPrzxZ4j8c25i8BW62tgTsm1fUUKDPpCvUkddxHXjjrQB03jDxrpfhGxSa6Zri6mIW2s4cNLMc/wj09/avNvC/hfVvH1vrr6tqF7o+my6tK1xpEAAdpCsZIeU8kY28FTXo/hfwHpPhkfaI0e81OTmbUrs+ZPIe5yfug+g9ar+BQBqHi5eh/t2QkccZiiPagDpNK0my0XT4bHTrWO2tol2rHGuB9T6n3JzV4dKUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUn40Acr8RNWOj+BdVuE5meE28I7mSQiNf1bP4VreHdHh8P+HtP0i3OY7OFYs46kDk/icn8a4Px20niH4meFPC0HMNtJ/a16P4dqH5AfqQw/wCBCvUBQAtUNY0u11rTZ9OvrdLi1uEKSRv0I+vbp1q/RQBz/hbQn8MaFBo7X8l5Fbllt3lUK6xZ+VDjqVHGcD6Ct8YHGaztd0tdZ0a805ppYBcwtF5sLFXTIOCCPSuM8G65q2i6jH4P8XyA6gFJ0+/LZS+jXrg/3wMZB5OaAPRD0NcNppj0n4vavZDAXV9Ohvh7SRM0bfmCpruBwpzj6V5j8SpX0Hxj4M8T8pbQXb2V0w/uSgYz7YDfpQB6eDx1rL8SaumgeHdR1aQAraWzzbT/ABEDIH4nitNeleZ/GKWXU7XQvCNq7CfW75Vfb2hTlz+GVP4GgDpfh7pI0fwJo9tj948AuJT6vJ87Z/FiPwrqaZEixRLGgARQAoHQAdKfQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABXOeO9e/4RnwVqurLjzYYCIgT1dvlX9SD+FdETXnXxPtJNf1Dwp4aTJS81H7RcL2MMKlnB/76H44oA3vh3praV8PtDtXBEgtVkcN1DP85z+LGuopiBUQKoAA4AHb2rF8S+K9G8JWAvNZvVtombanyl2dvQKBk4/TvQBtFwDzxXEeMPiNb6BqUOjaXp82s67N92ztiD5fu552/THT0rJul8XfEKSJbcSeHvC82C0hfbeXMfpgZ2A/Uceua7Tw74Y0nwxZfY9JsYoIyPnfGXkPq7dWP1oA5OP4f3/ia9tdT8bam12Im8xdHthttIj/AHTzl8cc8c56ivQoLeOCBIY0VEQBVVRgADsAOB+FSqMCloASuO8EDGseMh6a2x/8gxV2Jrj/AAaNviHxmvpq4P5wRGgDsaKB0ooAKKKKACiiigAooooAKKKKACiiigAprE9uvpTqwvGWux+GvCWp6vIebeE7Md3Pyr/48RQBz/hO3j1bx/4q8SDlI5V0m374ESgyEfV2/Su8FcZ8LNIn0X4d6XDeZ+1zhrqYnqWkJYZ99pXNdpQAUUUUAFYXiPwrpniYWI1CN2ayuVuYHjbaVcdsjse/0FbtIaAM8axYf2udJNzEL9YfP+zbvm8vcV3fTIxXM/FXRv7d+HGrRR5M9tH9qiI/vR/Nx+GfzqXxx4UbWrWPU9K8q38Q6ewmsbo8HIOSjHupGRg8c1Y8JeJf+Eq0q5W70+azvbaU219azLwkmMkA/wASnIwfQ0AbOi339oaBp1/2ubWObP8AvKCP51yb20et/GZbhvmj0DTQAD2nnLD/ANAX9RUXwu1dE8By2lxJtfQ7ieznLdhGxI/8dIH4VB8JJJtTsdd8SzoR/bGpySwbuphQbVH4YI/CgD0helLSDpS0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFIaWkJxigDm9c8deHfDd4tpq+o/ZJ3GVDQSEN9CFI7inWXjXw9f3i2kWppHcOcRxXCPA0nsocLuPsM1478WdQg1H4x6Bpt5cxW9jYeU88szBUXc25yc/wCyF/KtLxpcL8WvEWj6X4UzdWunSGW81FoysMeduAGOCTgHgeo9DQB7gTkHmvPNO1OPV/i/rczSILTQtP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proof	Solid Geometry	Math	Chinese
44	"如图, 在凸四边形 $A B C D$ 中, $M$ 为边 $A B$ 的中点, 且 $M C=M D$. 分别过点 $C, D$ 作边 $B C, A D$ 的垂线, 设两条垂线的交点为 $P$, 过点 $P$ 作 $P Q \perp A B$ 于 $Q$. 求证: $\angle P Q C=\angle P Q D$.
<image_1>"	['联结 $P A, P B$, 分别取 $P A, P B$ 的中点 $E, F$, 联结 $E M, E D, F M, F C$, 则四边形 $P E M F$ 为平行四边形.\n\n从而 $\\angle P E M=\\angle P F M$, 由 $M E=\\frac{1}{2} B P=C F, M F=\\frac{1}{2} A P=D E, M D=M C$, 所以\n\n即 $\\angle D E M=\\angle M F C$, 所以 $\\angle P E D=\\angle D E M-\\angle P E M=\\angle M F C-\\angle P F M=\\angle P F C$.\n\n又 $\\angle P E D=2 \\angle P A D, \\angle P F C=2 \\angle P B C$, 得 $\\angle P A D=\\angle P B C$.\n\n由于 $\\angle P Q A=\\angle P D A=90^{\\circ}, \\angle P Q B=\\angle P C B=90^{\\circ}$, 则 $P, Q, A, D$ 和 $P, Q, B, C$ 分别四点共圆. \n\n故 $\\angle P Q D=\\angle P A D, \\angle P Q C=\\angle P B C$, 所以 $\\angle P Q C=\\angle P Q D$.']			['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proof	Plane Geometry	Math	Chinese
45	"如图, $\odot O$ 切 $A B, A C$ 于点 $B, C$, 过 $C$ 的割线 $C D \| A B$ 交 $\odot O$ 于点 $D, E$ 是 $A B$延长线上一点, 直线 $C E$ 分别交 $B D$ 和 $\odot O$ 于点 $F, G$, 延长 $B G$ 与 $C D$ 的延长线相交于点 $P$. 求证: $A, F, P$三点共线.

<image_1>"	['连结 $A F, F P, B C$, 如下图.\n\n<img_4137>\n\n\n由弦切角性质, 有 $\\angle A B C=\\angle A C B=\\angle B D C=\\angle B C D$, 则 $\\triangle A B C$ 与 $\\triangle B C D$ 均为等腰三角形且相似.因此可得 $\\frac{A B}{B C}=\\frac{B C}{C D}$, 于是有\n\n$$\nA B \\cdot C D=B C^{2}\\cdots(1)\n$$\n\n又由\n\n$$\n\\angle B D P=\\pi-\\angle B D C=\\angle C B E, \\angle D B P=\\angle D C G=\\angle B E C\n$$\n\n可知 $\\triangle B D P \\sim \\triangle E B C$, 则有 $\\frac{B C}{B E}=\\frac{D P}{B D}$, 从而由 $B C=B D$ 可得 $\\frac{B C}{B E}=\\frac{D P}{B C}$, 于是\n\n$$\nB C^{2}=B E \\cdot D P\\cdots(2)\n$$\n\n由(1)与(2)可得 $A B \\cdot C D=B E \\cdot D P$, 故\n\n$$\n\\frac{C D}{B E}=\\frac{D P}{A B}\\cdots(3)\n$$\n\n由 $C P \\| A E$ 可知 $\\triangle C D F \\sim \\triangle E B F$, 从而 $\\frac{D F}{F B}=\\frac{C D}{B E}$, 联合 (3)式, 易知\n\n$$\n\\frac{D F}{F B}=\\frac{D P}{A B}\\cdots(4)\n$$\n\n由 $C P \\| A E$ 可得 $\\angle F D P=\\angle F B A$, 从而由(4)式可知 $\\triangle A F B \\sim \\triangle P F D, \\angle A F B=\\angle P F D$, 故 $A 、 F 、 P$三点共线.']			['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proof	Plane Geometry	Math	Chinese
46	"如图, 在 $\triangle \mathrm{ABC}$ 中, $\angle \mathrm{C}=90^{\circ}, \angle \mathrm{ABC}$ 的平分线交 $\mathrm{AC}$ 于点 $\mathrm{E}$, 过点 $\mathrm{E}$ 作 $\mathrm{BE}$的垂线交 $A B$ 于点 $F, \odot O$ 是 $\triangle B E F$ 的外接圆, $\odot O$ 交 $B C$ 于点 $D$.

<image_1>
求证: $\mathrm{AC}$ 是 $\odot O$ 的切线."	['如图, 连接 OE.\n\n$\\because B E$ 平分 $\\angle A B C$,\n\n$\\therefore \\angle \\mathrm{CBE}=\\angle \\mathrm{OBE}$,\n\n$\\because O B=O E$,\n\n$\\therefore \\angle O B E=\\angle O E B$,\n\n$\\therefore \\angle \\mathrm{OEB}=\\angle \\mathrm{CBE}$,\n\n$\\therefore \\mathrm{OE} / / \\mathrm{BC}$,\n\n$\\therefore \\angle A E O=\\angle C=90^{\\circ}$,\n\n$\\therefore \\mathrm{AC}$ 是 $\\odot \\mathrm{O}$ 的切线;']			['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAqMC3QMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APbaQ0ZozWpsFFGaM0DuFFGaM0BcKKM0ZoC4UUZozQFwoozRmgLhRRmigLhRRRQFwooooC4UUUUBcKKKKAuFFFFAXCiiigLhRRRQFwooooC4UUUUBcKKKKAuFFFFAXCiiigLhRRRQFwooooC4UUUUBcKKKKAuMPWikJ5oyDQO4tFFFAXCiik6CgLi0UmaM0BcWijNJQFxaKSigVxaKSigLi0UlFAXFoozRmgLhRRmjNA7hRRmjNAXEopCaM0Ei0UmaM0CFopM0ZoAWikzRmgBaKTNGaAFopM0CgBaKKKACiiigAHWnU2lyaAFopMmjJoAWikyaMmgBaKTJoyaAFooooAKKKKACiiigAooooAKKKKACiiigAp4plPFABRRRQAUUUUAFFFFABRRRQAUUUhoAOKOKbS5oAXijikzRmgBeKOKTNGaAF4o4pM0ZoAXijikzRmgBeKOKTNGaAF4o4pM0ZoAXijikzRmgBeKOKTNGaAF4o4pM0ZoAXijikzRmgCFgd+aUA1OEGKCoHSgCOnFgKa2KryOQaAJi4HemSSqqkkiqNzdCFSzHgc1y2o+J0ab7NA4aU9Ae9DYrnY/ao/7y0ouY/7wrz55teZQ0cIP/AqhvdT1qyRDLFtz15pXDmPSlmVuhBqXcDXG+GtVlvk/e4zmuuFNDRIOlLSA0uaACijNGaACijNGaAENJS8UlABRRRQAUUUUANNFFFABRRRQAUUUUAFFFFABRRRQAU8UynigBaKKKACiiigAopKM0ALRSZozQAtFJmjNAC0UmaM0APooooAKKKKACiiigAooooAKKKKACiiigAp4plPFABRRRQAUUUUAFFFFABRRRQAUhpaQ0ANoozRmgAoozRmgAoozRmgAoozRmgAoozRmgAoozRmgAoozRmgAoozRmgAoozRmgAoozRmgAoozRmgAEuOKRpAagZvmoDUASM+aqTNtUt6CrNY+vTm3sWcehoA4fxl4qW1/cRsdzjbxSeBNL+3x/bLtdzg8GuBlujq/iRYn5xLjmvddD01bO1SNABlRUrVkrU1EjjRdoHFZWt6at+ihQOK2fKNHle1UUc9o2j/AGDqB1roc0nl4oIoAXd704EkVHT1pgOopKKQC0UlFAC0tNp1ABRRRQAUUUUANOKM0jUgpgOopBS0gCilxSUAFFFFABRRRQAU4Gm0ooAfxRxTaKAHcUcU2igAoopcUAJRS4oxQAlFLiigBKKKKAJKKXFGKAEop22kxQAlFKRSUAFFFKBQAlFLijFACUUuKMUAJThTSMUuTQA6im5NGTQA6im5NGTQA6im5NGTQA6im5NGTQA6kNJk0hagBtFN3UZNADqKBQKACilxRigBKKXFGKAEopcUYoASilxSUAFFFFABRRRQAUUh4oBoAWiig/WgAopB1607GRQBEy5NAWpcCgKKAGYrnvFiltKbAycGulK8Vn6laC6gaMjPFAHzboitF4sO8cedX0nZOpiiIIwFH8q8P8S6DPpWrJcwrgeZuOK7nw14lEtgBMx3DjmpWhK0PR/NjHORUX2mAttDDIrmm1uN12o3NFgZpZmY9DTuO5025WFIQDVePIxmpc0xi7fSgDFG+jrTAKKXFGKQCUUuKMUAJTqTFGTQAtFJk0ZNAC0UmTRk0AMagUGgUwFFLSClpALSUtFACUUtFACUUtFACUoooFAC0UUUAFFFFAAOtOpo606gAooooAKDRQaAG0UUUAT0UUUAL2pKXtSUAIaQ0ppDQAlOFNpwoAWiiigAooooAQ9KbTj0ptABRRRQAUUUUAFFFFABRRRQAUhpaQ0AR0UuKMUAOFKKSigB1FJmjNAC0UmaM0ALRSZozQAp6U2lJpuaAFopM0ZoAWikzRmgBGpN2BQxqKRsIcdaAHmZRwWApjXCAfeFcB4p8UTaTI4A6etVNL8RahqUe6OIt3+WlcVz0cXSZ4IzTxc158muXdq5N2nlqOpPFb2na5a6ioMMobPpyKLhc6lZMqDQZMVDC2Yl70r0xknmimlgc+9Q04GgCjqGj2+oIQ8ak9iawf8AhCmi/wBU+wZzxXYIwHU0y8vYbaFnkYBR3xQBy9t4Xlik3NNkV0VvaiFAoAz3qO21GC5QNG4INWhICOtCBDtlP20zdS7qAF20qjFJuFKGzQAtFFFABRRRQAhpKU0lABRRRQAUUUUANNAoNApgKKWkFLSAWlFJSigAxRinUUANxRinUUANIopTSUAFFFFABRRRQADrTqaOtOoAKKKKACg0UGgBtFFFAE9FFFAC9qSl7UlACGkNKaQ0AJThTacKAFooooAKKKKAEPSm049KbQAUUUUAFFFFABRRRQAUUUUAFIaWigBtFLigigBKKKTNAC8UcU3JoyaAHUUZozQAUUZozQAHpTOKcTxTKAF4o4pKKAF4o4pKKAEY8VVmk8uIt6VbYZFZOrv5VhK2cYFAHl/i/bf6nJE3INd/4W0a1stMheJDuZec157HG19rh5yDXrOkp5djEhHQUkSjN1/QYdSs5FK5civJNOurjw34vXTi22Idvxr3mYqIiTgcV4F4tjM3xEKxtyQefxpSCR7ppk63FpHIvORV0qCOaxfDaNHpMAY5OK2d+KopCFBULnaTUpcVl6nqMdlC0jkfnQBLc3fkRsxPAFeWeNPF07s9pbSfM3aoNa8U3Gq3v2a0LqN20kVvaD4REqrJeqJHznJqdydzlfBfii8i1AW94+EGMV7NYTLPCrqchhXB+K/CkdrZfaLKMJJk8r1rP8KeLWtbj7LdZ+U4+ahaBseuKoNPKCqtlex3EasuOQKu71PaqKK5HNSIMCpdoI6U0gDpTASiiikAUUUUAFGBRS4oATAowKXFGKAEwKMClxRigCMj0pMGpKKAGgYop1GBQAlKKKBQA6iiigAooooAKbg06nUDQzFGKfRQOwzFGKfRQFhmKdilooCwmKMUtFAWExRiloNAWGYoxTqKAsSZo4pKKAsOzSZpKKAsBpKWigLDeaUGlptAWH5ozTKKAsPzRmmUUBYcTxTaKKBMKKKKBBRRRQAZozTaKAHZozTaKAHZozTaKAHZpM0lFABTadRgUAMop2BTGHpQA7NIDSDNKBQAtLiil4oAbim8+lScUcUAMxSU449abmlcBM0uRUZBzxTTmncCYEEHmuQ8Y6ilrZSxlsEjiumkk8uNifSvI/Hl81xeBFPGMUmxMk8HAXOpJKeQa9UgdY0Az0rzDwXstoI3c4Irr59ZhVDsfkCkmJF3xDq8VlpsreYAwXoa8b0qK413xgl75ZKHuK6DWGv9XuREozG3BxXaeDvDUOmWSFlxIPajcNzodOhNvZxoRjFWHlA6mnzOiKMHGKwtX1WG0jJZsNjincdyTU9bi0+FnZ1BHrXl2s+Jn1m6a2jPBI5U0a5f3GsTtHAdyt71hw6NdabOLllxzzmk2Js9C8M+E0hUXD8lufmrura2VCFGB9K5jw34it7i3WFmG5RjArrYHDOD2oTC4l3axTIY3AI968p8VeG/s1wbmAkYbcQK9UuphGzE9K4DxNrMLI0KNlm4psGVfCHinkQSsA2ccmvSYbwSLuHQ18+Q2V7aXf2pF+QHOa9H8M+J1lVIZn+b0pJgmekJMCKXeGOaz4pPMQMp4IqzHnbVJjTJ8ijIqEg0oBoGS0EgdaUDNRyA4oAXeDxmnjpVRQ2/OKtjoKYC0UUUgCiiigBtFFFABRRRQAUCigUAOooooAKKKKACnU2nUDQUUUUFBRRRQAUUUUAFFFFABQaKKAEooooAdk0ZNNooAdk0ZNNooAdk0ZNNooAdk0lJRQAtFFFABRRRQAUUUUEsKKKKBBRRRQA2iiigAooooAKKKKACiiigAooooAKTilpp4oAXAoxTM0uaAHUc03d60u4etAAelRFiDUpYYqBjzQAu73pw6VCWApvnqP4sUguWaXiqhukHO4Uw30f98UhXJbtN8ZUd65O88I/bZt7KD9a37jU4IgC0gH1quNfswf8AXpQBnW3hY28e1cYp8fhmQy5JUg1cfxDa9pUpF8R2yn/WpijQNC7a+HreBdxjG71q+qCMbQcCsr/hJ7PH+uSs7VfF1pb2rSiVOKLhoaOq6hFZwszsue3NeYX97daxqAjhY7N208VnXniG48Qag0EZYICORXf+G9ItrSNXkYMxHU0tydyXQfCqwxCSVATWpqegQT2pUR84rZiliAAUgCpyUcdQfwoKPC9UtLzQtRV4ywTfn5R2rufDniVbm2G5vm6c1t67okN9A2VXOK8k1qO70C98uBXdc54NFrE7Hd+KPEaxWpSJvnz/AAmub0HSLnVrtp58lScjNHhzQ7vXJlubhXVG7GvVdM0iOzhVEQAgY4p2HYzR4bt5LMxmMZIrz/XtEn0W6aeHhR0wK9lEBUdKydY0tb6BkZBk+tFgscd4Z8UpMoglOGHHPrXeWs6yx7sjkV47q2hXekXpniD7S2eK7Dwt4iW4h8uQgOOMGhaAju+DTgKqRTB2AzVkE4qiiQU7ANQbyDTlemBKYwBTaXeCKSgAooopAFFFFADaKKKACiiigAoFFAoAdRRRQAUUUUAFOptOzQNBRRxRxQUFFHFHFABRRxRQAUUUUAFFFFABRSjFHFADaKKKACiiigAooooAKKKKAFooooAKKKKACiiiglhRRRQIKKKKAG0UUUAFFFFABRRRQAUUUUAFFFFABSEGlqKRiPagBxBpvQ1E0uOrAD3NU7rUIraJpHkXC9s0XC5eZwMmoTcIo5PSvOtb+JFnbStCiuW9RXO/8JTqeruUs2dSfVTRzC5j1mfWraH70mMVk3PjDTrcnfN+tcLa+GvE903mS3IKH2NbVp4JmkYfawrjvxU3Yrslu/iHp+SsU+W+lZsvjKeUfuGDenNdVD4C0FVBezG7ueKvR+EdGiHy2+PxFGoWPPpNd12a3ZoIwT25qKCbxlcR70tFKn3P+Feqw6HYRrhIeKvw28cKhUXgdqLBynjNzp/jG8GHtPyY/wCFV4fDXiffmW2I/wCBH/CvdkUZ9KeyIRyBRYdjxdfDOtkcwtn60ybw1rSoT5TdPWvYJiqE8gVyviTxDFYW5GctjHFHKhcp5DrLanpnEgKt6ZqhaWOueILQLHGWU+hNdhpGnXHiq/8ANm+aJXI5HavV9I8PWWlwKkMW3FKwkrnh+neE/EmlMJI7QH6k/wCFdHFfa7bp+8iAx1r2Iwxuu0iqz6RZSAh4+vWnyj5TyxPF09sP9IYKR71p2PxEsA22efB9q7Kfwhos+d9sDn3FZdx4A8PkHbZgH8Kdh2K0XjTTLoYW4Boaz0vWZA7tn6CqF14FjRW+yRhD2rnrnwp4qtyTa3Soo9v/AK9K7EepadHa2FssEONq9CRWnFcp1zXg8mr6/oL4v5mkC9dqk1u6V8SbZmWKVHDHg5FFwuexCdT0ob5+1crYazHeIHSQAH3rXiuWOMOD+NMYzU9JF7C6lB09K8q1nTLrw7fieJcRA5YmvZY5WI61nappltqULJOm7PWiwWOX8KeJrfVFjTzAXPau5XlfrXh/iPTrnwpqL31l+7tk6ADmu38JeM4tYt0hLHzVHzbvWhPoCfQ7dkJ6U3aaSOTcM54NTrg0xjApFPpxAxTaACiiigAooooATFAWlozQAbaXaKTNGTQAbaMCjNGaACiiigAooooAKKKKCkLRRRQMKKKKAClpKWgAooooAKKKKACiiigBKKWigBKKWigBKKWigBKKWigAooooAKKKKACiiiglhRRRQIKKKKAGHijNKRmkK8UAJupQaZzSjrQA+igUuOKAGg80tNICmmPMFFAEh4FQtcbc1Xn1GOJNztgCuY1HxVbRbhHIpYdqL2C50k2rxQDLlRiue1Txhbw5CFWPoDXFX2qanqblY4yVJ6g0aV4Rubu4D3AcA+5qbtk3bLl54tvb393DbSDPGQKrL4V1bVQZGvpkU9s13uneGba1jUFBkf7NbsVuka4VR+VOw7HnWlfD1ItpuXErDuwrr7Hw3ZWoBWCPPritsIPQU/bTsg5SFLaNAAEAA9KmEa4+6KXFOFAxuwelHl0/NGaQEZAHSmk4p+KYyZoGML0SXGxCWOAKZOwhXcTwBXEeJfFiW0bQQuC57Gi5LYvifxMLXckTbmHYGuUs9KvfEN1vlZ1TOeaZo2l3Ot34nuFO1vQmvV9L02Kzt0VVGQPSkJK5W0DRI9LhwAMnmt8t6CkEeO1PWM4plEOwls7jT1UingYNLVDEyfWmsuafRQBD5QoEKk81NRQBi6j4fs70HfAhPuK4/VPh/DKSYNsZznIFek8VE4z2FKwrHjcnhzUtFYulzK6jtU1t4yvbBhHLayEDuRXqk9pHOPmRfyrn9S8MW91EwCLn/dpWYmmV9K8XwXgUO6xk9ia6OK/iljwrK+a8ov8Awjc2UxkgDnnPU1JZaxqemMEeP5R3JpXaFdo9B1jSotWtGhdVOfUV5TdaXdeFNUaeMuUdgMCvQ9L8VW8xXz5FDdxVnW7O11mxO3BwpwQKb8g3HeG9ZS/tl/eDcO1dOHKjPWvDra5vPDuoEEERs3JJr03RPEMWowD5wWNCYJnSC4zxipN+RVOMAnIyc1ZB4qiiTdRupuaKQD80Cm8UoNADjTCadUMjhcknFADifekB561Sk1C3TOX4qFdXtCceaKLhc1fxpRzWX/bVlu2+byeK0YpBIoKng80BcmooooAKKKKACiiigpC0UUUDCiiigApaSloAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKCWFFFFAgooooAAaD04ppODSryaAGY9qUKRU4XIpHXFAEQ4pS4AqORtoNZl5fpbxlmYce9AF2WdR1PSsLVtdt7JSGOT7VyOs+MCsrRxZLE44rPsNNvtfk3yNIin1pXJuRan4iu9RumitXO09K0dM8K3F0FmmGdxya6rR/B8NiVMgVyO5FdNHbJEm1VAHtQkNIxNN8P2ttHzHzW1DbRRfcXpTiMVJF1pjF20uKk20hoAZinYpaKAE5pKdxQQKGAw4FGRTXOBVZpcZpAW94qKa4SFCzEYqlLdrEu5iBiuA8UeKHLPbW5JYf3TQ2K5e8S+LVyYIGOSccGsDTfD8+rXAnmGRnPNL4e8N3GpXP2i4ZiCc/NXqVlpsdpCEVQOO1SKxQ0rSYrCJVVMYrZjGDSiPHpTgMU7lEq9uamXAFVgcU7fQArYzScUhJNHNaJFWF4o4pOaOaLBYXikJo5pposFhaSilFFgsNIpmwmpcU7bRYLFGW0jlHzisbUfDlvcIcRjP0rqNtBRcdBQwZ5Lq/ha4s4jNAMEVi2niG9sJhFO52jjivari2jlj2soI965PVfCUF2WdEVTUNENHO3Qs9XtdwHz47+tc1BcXGh6gF3YQVpXmlXmmOxUuUHoKybm5W6jIkG1/U1LJZ6noOtx3kCZb5sc810atlcjvXg2kazJpt6AWOzIA5r2HRdVivrZSGGQKpMaZtA07NRA55zTgeaook4peKaMUtIB/as7Um2QOw9K0M8VgeIroQWbH/AGTQ2Js8/vdanl1lLVH4ckV1Fto820Mw5rh9KaO81+OUkAK5/nXptzqEVtGWBB/GpQkcF4it77T7qB4zgNIM/TNenaKzNYRFuuwZqhNYxarbwyMoPORmti1j8mMIMDAwKaGi1RSZpc1QwopBS0WAKKOaKBoWikopDuLRSUUBcWlpo606gLhRRRQFwooooC4UUUUBcKKKKBhRRRQAUUUUAFFFFABRRRQAUUUUAFFFFBLCiiigQUUUUAMYZ7UoODk1KoBByajf7tADvOVR1qN5xjPpVCV2DH0rG1nWU0+2cu2GI4+tDYNmjqeqxWkLsz4x615tquu3Op3TW9su/PZTzWVPqeqa/qPlQNuhY969C8OeFIbSNJpo/wB93NTuTuYeh+BvPkFxdF1Y/Ngk131rZR2iBERfwFaEaKsagcYGKXaDVFWIlOKkBpdi0baYCbN1OVdtA4oB5pAPJpM0hpKAFzSbqaSaaeKAJcigmoqCfU0MBsp4rOurlbeNncgAc1PeXSW8RdzxXl/ibxeLmY2do5LMdhqWxMn8QeJpJ5WtrTDsc4APNR+G/Ctzc3YvLyNgzdQc1f8ABPhMsn22+jzKGyPpXptvbxooAGBRYEjP0+yS2QIFHHtWlgYp8iBRxUVMYtIc0DrTqAI+fSk5qYAUu0UARiloPWirRYUUUUwCkNLSUAJS0lFADs0u6mZozQBJkYpjPikzShQaTExud1MMWamwB0pCaRJQudOinjZGRefavOfE/gx1maa0VmPYCvUyajkAYYPNTYGj5m1mC7suJYimD1NaXhPxpJYTiF2+UnGSa9Z8T+FLTUrZiI8vya8F1/QZ9IviduF3ZBx1qWrENWPo7RtXi1K2WRHDZx0rXBrwvwD4rS0eO1lcjJr2yzukuoVljOVNUmUmXAaUUwfSnimMU9CTXCeONQRLUxhvmIxXZ3E3lI3PGDXifjK+luNTWNWGPMGRUshmf4esdSMr3EMDsA5P610kMuoalqv2KeMqvt1rp/B8Cw6aVI+8BW0lhaQXf2vGJBSQGjp1t5FjFGc5UVeC4HFYGmeJbTUNQms4mJkjODW+GyKtDQppc02lFUUOGKcMUwUtMB9Npc03NJjFopMmjJqRC0UmTRk0AKOtOpgNOyaAFopMmjJoAWikyaMmgBaKQUtABRRRQWFFFFABRRRQAUUUUAFFFFABRRRQAUUUUEsKWkozQIUClwKbuo3UARykqQAetR7iOCafLyR6CuR8T+KotHjdVZTIOimhuwN2JfE/iCDSrUsPmY8YBrzyz+3eJr8/M4jDdx2otYLvxRfl5VdY2ORzxXqWhaHDp0CgKu7HPFTuTa5Donhm306JSY1LgZzXRBAOAOKdgDsKDVFCUUUUAFFFFABSA4NLTWoAk4NTLtx0qqHqRZMUASvsA+7UEjADpQ8vFQPJ8tACPIADWTqGqJZRl3ccds03U9XhsImZ3CkDvXm2ralPr90YYMlW7qaTYmyXXPEtxq0xtLUsvPUelbPhrwUIiLm8USM/zDParHhjweLULPLlm9GruooRGiqAMChIEhbaBYItiAAdMCrCMR0NMAxThxTGPYk9abRmjNABRRmjNABS5NJmjNACjmlxTQadmmMMUYozRmgAxSEClzTaADApDS0hoAaaSlNJQAU5WxTaPxoQmOLimb6DSYpgLy1J5eaUE04GgCNoNwIPOawdY8M2WoQt5kClscE10e4VG2KkR84+LvDU+i6gbi2BVV5worf8CeOzFKltdMTjH3q9P8QaPDqVq8bKuT3xXh2v+FrnSr5p4A+0HjFQ0S0fRFpex3UKSIRhhnip99eOeEvGrQ7LWZgCPl5r1azvUuog6sDkCmmNMxvFOqtaWbbImY89K8NvdSupNVaRrSVgr5+4a+jp7WO4TDqGz6is5vD1kTkwRZ/3RRYTPOdM+IIs7dUOmzZA/uNUerfEGa9tjFb2U8bnvsNek/8ACP2X/PCL/vgU5fD9lu/1Ef8A3zRYLGH4L0d4UXUHGHmGTkc126ZxTIIUhjWNVAA4GKkbAqkNDs0ZpoxS00UOU06mCnA1QCmm8UpptJjY7NGabRUiHZozTaKAFB5p9MHWn5oAKKM0ZoAKKM0ZoAUUtIvSloAKKKKCwooooAKKKKACiiigAooooAKKKKAClpKdigljCaaTTmpjUCFyKY0gUZJ4HNNLY5zWLrerR2Vq+WG7aR1oYFXxD4nt7CBkD4kPSvNksbvxXqqzSDdET1pkNre+JNXDbm8pX5z6V65oegw6ZbKgjww7ipepL1ItA0GHTLVFC8j1rfUAdKNvtS4qihaQ0tIaAEooooAKKKKACmnNOpKAG80vNLig4FAEbdKxdX1aGxhJZsH3qfWdYh023Z2IyPevKbnUrjX9WaKItsBHbik2Jsm1Ka71662R/MmccV1nhTwoLJUklTBrR8P+H47eFXkQbjzXWRRqiYAAoSBIasSouB0FG2pDTaYxKQ0tIaAGmlzSGkoAdmjNNooAdmjNNooAeKWmjpTuKACijijigAoo4o4oAKQ0tIaAGmkpTSUAB6U3cAadUZjzTQC7ge9KDTdhFIWx3qhDzQTiq7zYqCW7AzlgKlsC7vqNm4rDutcitslmB/GuS1fxqFfZDuyeOKlsVzuru+hgB8w1xXiXVtNnt3jVsvg9q5/7fqWpyfK0gU/7Nadl4SublmkmbII6GluG54/dXD22qtKhwN+RXrngXxak0KRTS/NnGM1w3jfRl06RcAZJ5wK5XT9TmsLhXRyAPSp2J2PrWCdJk3Kc5qUmvKvBfjIXKxxSyc98mvULeVZ4w4IIPNUmMcTQHIqTZmkMfFMYCSlDbqZjFOWmCH04U2nCqKAU4U0U4UxhSUtJSYBRRRUiCiiigAHWnUg607FACUUuKMUAJRS4oxQAq9KdTelLmgBaKKKCwooooAKKKKACiiigAooooAKKKKAAU+mU4mglkb0xzxSuwFVLq6SCFnbsKBGfrWqJp9szlhkV5Rc6heeINUeBFfYD1FaninU31O5e2tzkt2zXTeDvDS2cCXDJ87DnIqdydzW8K6HFp1qCygueeldMABUUKGMYqWqKCiiigApDS0hoASig0meaAFooFO2nFADajdttSE461WncAE0APMmASaxdZ8QQadC26RQQOM1X1nXIbKA5cA9q8wu21HxLqAWMbot2CQe1JsTZZvry98TX5jiV/KfuprvPDHhRLCJZZMFiOc1L4U8NR6TZoGX5/cV1qKB0AFCQJAkSqoAAAFG7FSY4qEjmmMdvNGabRQA6ikyaMmgBGFNpx5puDQAUUc0hoAWikyaM0AOFLTQaUUALRS4oxQAlFBpu73oAkpDSBqOtACGkpW4FQtKFFAEvFG5QOTVR76JM5NZt7rVtCpLSAAUXC5rTXKRoWLAVk3OrwRZy4rjdc8bW6wvHBLmQdAa4uS/13UZP3UeUPQ5ocibnpN74st4jgSLntzWBf+Irm5J+zxs+f7tZukeFLu8YPfKV+ma9B0fwvZ2oGASfep1YXZ5+lnqepSYkilQHvXQaf4GDYkllJOc816IthBGowo49qVY1B4xTsBi2OiW9ogHlqce1aBgVEbaoHFX1QYpsqgRtj0oGeE/EePzLyKMdWbFcPNorxzKhBG7mu/8AHYzrVqO3mVHrloIRHcqBhUHOPaoJOKhS50iQTJuwK9d8F+L47iJYpXAYDGCa4v7OuqaaNoBbFcglxcaNqRI4G6gR9UQTLKm5WyCKlPNeeeCPFsd7bLHLJ8/QAV6Cj7hx3q0MMUKMU8UGmNCU4U2nCqKAU4U0U4UxhSUtJSYBRRRUiCiiigBV60+mL1p9ABRRRQAUUUUAFFFFADqKKKCwooooAKKKKACiiigAooooAKKKKACkJwetLUUh5oJZDO+ATngc1wHi/X/KJgifDMMcV0+u6rHYWzEsMlcYrzazs5tc1ZZGztD9/SkyWang7RXu5Y7y5Xd65Fep2saxRhUGAKpaXpaWNqI1A/CtNVAHSmkNAaSlNJQAUUUUAFIaWlwCaAImJpKsGLgUwx4oAjyaQuRnmnkVBKQoJY4A5oAgmmYHrwK5HxP4nTS4Hy2WHpSeI/EsVmGijcM59DXL6do9x4mullmLCNuxqdyWzO083/ijUDtZhEGz8w7V6vo2g2unwLsiw2OSBUujeHoNLhUIihgOorZEYUU0u40u41Bg1KKbjFOpjJO1QHrTz0phoASilxRigBKKXFGKAEp1NpcmgBDTcU+l20AR4FG1fSpdvtS7fagCDFOH0qTaM9KcEoAYCacD607bgVBI+w0AS7QetIVUCqcl1t7VAb/14xQFy8eM1VlkZScNWXe6/aWiktOgI7E1yN947fe0dvb+bz1UUriudzLeiNctIox6muZ1bxnaaeCG+Yj+6c1yM02qa6+As0Ib0q5p/gGcuJLi6aQejUrtiuzNvPHf9oSmK1jkVicZ2mp7Lw/r+qjzDP8Auz2IrvrDwrY26KTBGW9cVuwWCRLiNQo9hVKI7HC6f4FgVg17Ernua6a28P2VuqiOLAHTFbX2bA60eVjvT5UBXitYoxwuKtRxqvQAUBSKcOKbAkzUR4PFOBpGHNIAUn1psxPlt9KctNnH7tvpSA8Q8cf8hy0/6611d5pS3fhx3wCwUdvauU8b/wDIdtP+utem6JEs2lIjDIKjioQkeU6C4tdU+yyg7R2q74u8GNNaLd26gZGTVzxHYfYNZedFwM9q67TpU1bSREcErHigR8/WOpXujXoCMQVY9K908DeLV1O1igmb98cZJrzrxRo4sNTB8nK5zmq+nXcmlzjUYCSv9wUr2EfRIYkZ9adXIeGPFsOqQqjlUcdQTXWI4dcg5zzWiZSH04U2nCqKAU4U0U4UxhSUtJSYBRRRUiCiiigBV60+mL1p9ABRRRQAUUUUAFFFFADqKKKCwooooAKKKKACiiigAooooAKKKKACqtzKsSl26VZPSuR8basLLSmCNhwaGyZHCeK9Ykv78W8TbgrAda7LwbpfkW2915ODXHeGdFk1G/a5lG4Mc163Y2y20KKo7UkSkaCdKdjiowwAqUHIpjI2pKc/Wm0AFFFFABQvBoooAkeQACq8twAetJMxrOvJhGNxIAFAFt71EUljxXGeJfFsUMZhgkBdhtxWf4l8RhIGht2IkrnNB0O81W+M1x86lsipvcncl0nw/c61eC4uA2M9q9Y0bSI9OgVFUceop2kabFZQKoXBxzitXIqkhpAQKaaXNNJoGFIaM0lAC5NJRSigBKKdgUYFADaKdgUYFADDSc04iigBKXdTCaTIFAE24UbxUBYUhcDvQBPvxTTOFqCSVQDkgVj32tW9sDubP40XC5ttdjFZ93qsEGTK4FcZqfi+II3kbs9q4+51LVdUmKxudp/2TSuTc9A1HxXZR/KkwLVy1/4o1OVitpGGB96m0TwjPeHfOoY12dl4UtrYKXj5+lLVhY8sGk6nrFyWuVdc/wB0muw0LwXHbfO4Jz1zXax6ZBE2USr0UYQYp2GkZ1tpcUIG1B+VXRDt7D8qthaXaKYysBjtViFuOaUxZFRn5elNATMQaZxUYY5pcmmIWkIpaBzQAgFBp2KRhzUgIKbOP3TfQ09RzTLn/UP/ALpoA8T8YwvPrtqUGcS816foClbFAR2Fec3cgn15lbnbLXp+lLi1UewqEJHOeNrDzLFpFHPNYHgrUvJupbdz3wK9A1m0F1ZlOvFePys+k65nOAZBTYM7zxdoa31k8ipklR0FeeaLZoLwafc5VR3717FYzLqGmgk5+UZrzTxjZPpVy9/Fxg9aTEzKm0680LUPtFspMbNnk9q9Q8N6/He2yqz/ADAYOKzdLNrrugxqRmUIc59a46drrw/qQUHEbNk8dqExJntSyK2CDmplGa5Pw5rSX0KHdyfWurjYEA1XMVcdtPpRgipQ3FNbmqTKQ3FJin0YFDGMxRin4FGBSEMxRin4FGBQA0DmnUDrTsCgBtFOwKMCgBtFOwKMCgBtLS4FGKqwBRTttIRSKuJRRQKLCuFFLikosFwooopDuFFFFAXDijNNNMZiKVxXC4mWCBpGPC968h8RXjazrz2kZyh/xrs/GesrZaPPGGxIRkCuG8CWU2oXiX0i53d6TJbu7HpXhrS1s7KP5RuxW6wA6UyBdkKrjGKkPPWqKsR5NTxn1qLYKcOOlA7Ej9aZTS2TT+KAsJx60cetMYHPHSmb8daBWJqM1H5o9ar3N4sEZYsAKAJLqRIoyzMMDnmvNvFfifMptbb5ie60zxP4waVjbWzKzH5cA1U8J+FptQuRe3isGU9DU7kbi+HfD0+oOs9zuIbqGr0rT9LhtIlVY1BFXLbT4rePCqv4CrWzAppWKSIlGKUtQ3B4pAN1MAyaSnbMUmKAAUtGKKACnCm0u4UALRmkyD0oxRcBaKSlyM0AB96acetJLIqjrVN7oLyTRcLlhj71G7Ack1k3uuQWyks+MetcvqPjWJcpE4LdAMUriudrNeRxKSZAMVzd/wCMLe0mMeVJ+tcdNq2q6gxSCEsG9DRbeCLy/mE94JYye2TS5mLmuT6j4zmuHKQI/J7VTitNS1RwWMqg12uleArO2CyFyzehrqbewit1CqicdPlppDscLpvgh2jDSyZ9jXR2Xhq3tT/qlJ9cV0aKF6AflTqYWK1vAkIwqgVa+UjFRkHtQqnNAxfKFJtxUtIRQBHmnbvWgr6Um00APD00rvpuPapEOB700AzYRSbTUzEGm1QWI8GlHFOpMUgsLmmnrS4pjNg1Ihw60y5/1D/7pp6jjdUc53QSD/ZNIDxd/wDkYpf+uteraV/x7L9BXlL8eIpf+uteq6V/x7L9BUolGg0e9SMV5b400UpMk6jHzZ6V6ulYHiizFzaHAyQpNX0LexkeArv7RYSIx5Ukc/WrHjXRft+kyIFyx9q5bwte/wBmaqtoxxvc16rdqssBGMg+tQSeGeCNUksNduLOUnap2gHpXo2saDFqtn5qgbtgwa878T6TJo2si9jUgSygk/jXqnh7UI7ywjAIO1AKRJ5lDLc+HtSET7ti16ZomrpeW6ncMkVm+LPDy30UkqL857CuK07VJdDvPJl+VQQBk0AeyrLkDuKkVsisPSNTivoQyPuOOcVtRkFa0RaHmlHSkNKOlMoKKKKQgooooAB1p1NHWnUAFFFFABRRRQAUUUVaAfSNS0jVKAbSikpRTAKDRQaAEoooqQCl7UlL2oAYagmYKpYngc1Oay9ZmEOnzPnG1DUgeXeOb433iOCzQ5Rxg13vhHSE07T40C4x7V5/4dRda1v7U6hjHIRk9ua9jt41jiwBimhR3uS0UUUywooooAKUHmmk4NIDzzQBOFLCqlyQnU1Y3jBGcVl6lcLFGzswwvNAmyKa9jhQsx6VwniTxR5uba3Y78+tUfEXilpZmtbdDubPK1e8KeGnuJFvbtd4bswqdyL3HeFPBy3ErXeoR7gx3Ia9IsrWO1QJGMAUkEKRRKqqAAMYqygqiifNITSCigCJ+tLGcGhutMyQeKAJmOaZSCnqKAGjPpShDUgAxzTximBF5TYqLy2Bq3lQOtQO656ikwGKhXrTt4AqvNMqgneBWTdapDboxaZOO2aQrmu9wgNVJr+KI/McfjXA6n8RIbGUqsBl91rmrrXtR8SS7bWKe3DdDtpXFc9D1LxZZ23DNnFcnqnjdbg7LRmzVXSfAOqTS+bd3bOrHOGxXf2HhLTrZF8y2Rmx1xS1DVnmiaf4h1s5Q5RvUGui0bwCA2+/iyc54r0CKwt7f/Uxqn0FWVT15p2HYyrHQrKzZTHHgj1FbewFcHpSKAO1Shlx0pjIwoUYpaCeeKKYBRRRQAUUUUAOooooAKKKKACm96dTe9NDQDrS0g60tNjYUUUUhCGoXHzVK1RsMmgCVf8AUmoJP9VJ/un+VTjiLFV5f9TJ/un+VSSeMSE/8JJKP+mter6SP9GX6CvJZG/4qWT/AK6165pI/wBEX6CpRKNNKgvYllgkB54qccCkK71Iq+hfQ8S8SpJpWvw3KcKhzXqHhnVv7W0WO4Jzn1rm/HukedaySKOQvHFZHw51J4jFp8jEEY4NQyGdR4x0kahZbgM+Xlq57wJqqLJLAzfdbaPzr026thLYSjAOUOK8EufO0fxTGqEokkpPFIR7w22VPauD8WeFlu0MsSfNnNdjo0wubJSWBJrQktUdDuAOfWmho8X0HXJdFvBbXD4BbbXrml38V7biRD1rkPFfhBblWmtwEZeciuQ8P6zeeH9WjtLt3ZO5bpTWg9j28nNKOlUtPvor62WaNgQ3pV0dKssKKKKQgooooAB1p1NHWnUAFFFFABRRRQAUUUVaAfSNS0jVKAbSikpRTAKDRQaAEoooqQCl7UlL2oAjauN8a3fl2RizguMCuwlYIpYntXlXj3VBJf2UETZJkAYfjUsTLHw00g21tM7gktIzc+5r08dKwfD9oLWzj4xuUE/lmt8VSKiFFFFAwooooAjkJ3CkYkDNK65+Y9qy7/VrewjJmbGKBXsM1HUVs4zI7YrzrWvFVxqE32e3XKsduQad4g1yXWZDBaNkE/pWl4X8JMH86ePknNTuRuHhjwl5rrc3Gd2c4avRYLOO3QIqgAdMCn20CW8QVRUpbNUi0hNvtSqMU4DijGKBC5NHam0uTQAhpNmafgUZCigBAmKUnb2qNpwM1Tlvgp5PSgC7JMVFQPctjise71+2twfMbFc7e+MrckpDIS3ak2K510+orApMjbRjqa5fU/GVrbMVjnUt6Vyt7f61qrbUwUJ/SksPAtxdyCS5iGTSu2K7Y6/8Y6ldnbbwbwf7tV4tH1HW2Bm82MexNd/pfg+3tEUNH0roYdPihACDj6UajscFpXw9iRA0rsxP97JrtNN0C0solQRRkj/ZrVUBR0p24CnYLDPJjUYCgfhSeUKczigMKYxnl+9IUIqUc0pFAERJxTPMPpUrDioytADl5p1IKWgAooooAKKKKAHUUUUAFFFFABTe9OpvemhoB1paQdaWmxsKKKKQhrU1RmnPREOtACtwuKrTHEEn+6asydaqXP8Ax7yf7p/lUsk8VkY/8JNJ/wBdq9h0gf6In+6K8ZiSWfxRLgcCavatKQraqD6CpQkXTRk0pppq+hXQztWtFu7Z1buK8njibRfGXmDhB27da9llGUxXlnj61e0El2gwc9alks9Psbz7Rp6N2Za8r+ImmsmpQ3EacLySK7XwlqkN3o9uitmQLyKTxZYLdaVO5GWC1JJmeAr8z2UYJJOa77dxXkXgS5a0nSFjg5r1qNw8QYd6ECIplEq7SB+VcD4v8Mi5hkniBD9iK9AxzUVxAs0RUjINWWeS+F/FsmkXv9n3JAVMAEnmvXLS8S6t45FYEMM15V4v8GsJDeW0fz5zUnhbxY9o32S7fBX5RSvYV7HrgGRxTtgqlYX0V1EHQ5q7u44qihCopppS3pSDmgBB1p9Jt70tABRRRQAUUUUAFFFFWgHZoY5ptFSAUopKUUwCg0UGgBKKKQctUgKeKAeKV+lNA4oAytavFt7N29Aa8VsGk13xPKGywimyP0r0rxtcmKxYZx1Fcn8MbATateyuP4sgn6Cpe5L3PVrWDZBEBxhAP0qyTt4qZQAoAFV5+HNUihwbNKTio0NK5FAx24Ux5gvJ4qPdjk1z+uazFZREmQA+hoYmzT1PWYLOzldmXIGQM15DqWq3ev6iYrdmCE1LqN5e65ceXArlDwSprrvCPhdbKNHlGX96ncncd4Z8KeRGks6Amu6ggjgjAVQPpU0MKqigAACnMtUUROaaDzTmWkCjPNA7kwPApCcDrQQABzVeeXYO9AibIHem+YAazZL4KCWbFZN14hghzmVRj1ouFzpjdIvpVa41GGNSS6j8a4W98V+apWLk9sVz89xqt85EcUhB9DSuK52up+Lbe2Ugcn2NcfdeI7m/dhBvXJ9KdYeEru/kBuDIgPrXa6V4Qt7EA7g/rmlqxanD22lalqCfvHY59RW/p3gl8q8oU/Wu8jtoUGBEo+gqwqKvQAYqrDsZ1hoVrboMxjNaiW8Mf3VxS54pN1AwYCm5pSSabigAFLs3GjFFADZFwKYM1K3NN20AOWncU3pRQAh6VGetS03ZQA0dKdRtpcUAJRS4oxQAlFLikNADuKOKbRQA7ijim0UAO4ptFFNDQDrS0nelpsbCiigUhDX6U+DGDSFQarvKYzgUATy43VVuh/o8n+6amR/MGajucfZ5P900mSeR6PGp8RXWR/y1r1uyGIVx6CvK9GQf8JDd/wDXWvU7T/Uj6VKEiyaaadSEVXQoYRmub8VaUNR0549oya6U1DMgdMEZqWSzxXwdq0lj4hubN2IVGwM163cKLvTGB5DKK8f8SaVLomvG8VSBLIP516roN2l5p0YyD8ozUknn4xp/iRYhxj/GvV9Ok8yxib2ry7xbbtb+ImmUcYru/DF2ZrGJT1ApoaOg200ipB0pp61aLRVubdJ4irgEY715p4k8KtFKbm3Xbg54r1UpmqVzZpNG6MAc+tDBnmnhXxHLbyrbTMc5716da3QnjDBgc+leYeJ9BksLg3NsCSvOFqx4V8UOkiW9x8pHXNTsTsepLzipQoAqpZ3CTxK6NnPNWy3FUUBxSU3dlsU6gAooooAKKKKACiiirQBRRRUgFKKSlFMAoNFBoASmg/NTqjH3qkCYjOKXZhKQHimzSrHCSe1AHmfxDulSEoT/ABVo+AdN+yQG4xxIM1zPjq4W9umhTruH869H8PQCLRLUdD5fNT1JW5so+WFRzffNIpw3tSykFqpFDFyKbK2B9Kf0XJ7c1zHiLxJBp0LLnLH0NArkmu67FptszM+CK8rmu7/xHqbRx/NHuFXFN54pv9qE+U/qK9J0HwtbabboxjHm45NTuLci8O+G47C2VtmWbk8V08cG1umBTo/kGD0qQuoFUMeGCjBqN51HeoXnXJ5qpLcIOpGBQO5baYHvTWuAozWVLqMESljIg+prjtf8dQ2uY41ZiO6jNJsTZ3E+rRx5y2MVz2p+Lre3ViZOlefQ3Ws65cEQPIiuf4lPSuq0zwFfyYN9IsgPUUrtiu2YuoeLbjUAUsW3sfem2Ph7XdSYPNF8j+hr0ex8FaZa4It1DDuCK3oLdLdAqAAD0p2HY4/R/AltGA06kN3zXU22i2tmP3a5+orRUjvTjz70xlXylHQAfhRsIqztHpTcCgCID1p4pxApKAF4ptFFNAFFFFOwBRRRRYAoooosAUUUUWAKKKKLAJxS5ptFSA7NGabRQA7NJSUUAFFFFABRRRQAUlLSd6aGhR1paQdaWmxsM0mcUh4pjHFIRIWBqrNGS2RUiHJ61KwzQBFCCFwajuztt3J/umpwOcVW1LItH+hpEnl+inOv3Z/6aV6jaH90PpXlmhA/25df9dK9Ptd3lDr0qUJF0HNLzTU9xT+PWqKGGmEU8mkNSyWcZ490xr+xjKoCYzuNY/gK/LCWJz91iOvvXf6jCs1jMGHPlkV4jZ6hJoniLyM7RLIcVJJ2/jWH5pJx2q94Mn3qq+gqv4lIuvDrzA8+v4VX8CFvOIPpQB6MG4pMjNNXlaQ5DVaLRMnJpjjIIp8XU80n8VMZm3lgt0hVx1rzHxP4dn064a6tkPWvZwikdqz9UsIryApImaGgZwfhDxMD/o9w/wAyjbjNehJOsiBlPGK8d8RaRNo9159uCq7snFdF4Q8ULeRiGZsPnHNSnYlM9CTl81LUUJVk3DBzUowaooKKKKACikz6mjNAC0UmfejNUAtFFHFIApRSUCmAtBopDQAUz+Kn8U3HOakB2QBVO9fFu3NWX4WsrUJCIGFAHmeqxmXWmAXdyP516ppY26XbjGMJivOocPrzbh6V6TajFpGB6VKJRIxwCelVnlHJLYqW7YR2zOT0FcFrviZLeFo4HBk9DRcbZq+IPFkGnQFI3V3YbcA1wVtpt34mvxO7usYPeoLHRLvXr/zpw23dng16rpGkx6baKqryB6ULUQ3Q9Dg0uBVAUsO9bxIC9az3mCZJNZ95r9rbKTJLincZsTSYYfNioJLqOFcySBQPWuE1LxohbFpIGx71iXWsa1qqGNIgUPcGlcVzttT8U2dmpKTxu3pmuPvfHN9dM0UFg7A8blFGm+BpL5xJc+YCfc13Ol+E7SxjHygn3WjUDzeHRNW12ZXaWeAHtmux0X4frb7XuZfOP+1XaQ2cUX3VX8quKQBinYEinbaTZWyqqW8akdwK0FAAAxUXOeKkUmmUTVGRTgTS0AV2U/3qfCpB5NSbV9aQ7U70AOJpKZvFG/NAD6Q0gaigBKKXHFJTQBRRRVAFFFFABRRRQAUUUUAFFFFADaKQ9aMmpAWikyaMmkAtFJk0ZNAC0UUUAFFFFABTe9OpnemhoeOtOxTV6innpTY2ROcGq086RIWdtqjvU8lc94skMeiykZB9qTEyaTxDpsDEG7iyO2aaPGGm/wDPzF+deGx6fcapqsih3x2wxrfj8BXLoCPN6epqOZsjmZ6kPF+m+Z/x8xf99VBqXizTmtWC3MRJHrXmn/CCXIcDMn5ms/WfCNxYWwlBk6+pouwuzpvDN0kmsXLqAQz5FeqWtyvlD5O1ePeA4P37hicg17BaRR+WMk9KECLQbzOgxS+XSfKv3TS7z6UFB5dHl0bz6Ubz6UCGvAGRlJ4NeP8AxE0U2mqQXUS5C8nFexbz6Vz3i3TlvtKncgFlXikJo4Oy1H7f4YKN3NanhOQQXJGK5OzmXT9NMDHBB6GrvhvV2OpFARjIxQI9mtZAY8le1RzP8/SmWTZtlJ7rUjpubNMoakmDxUwHQ96jWLFP3GmhjxxSMc0mTTG3UwM7VdNivLdkZFztwOK8j1rRrrQ7/wC0228qpyQte0tvwRisfVNNF5Cyso5pNEswPB3i37ekdvONkh7Gu7jfdgjkGvFNR0+48P6sbuNSFHrXpXhvX7XULSNRKDKB8wHY0J2BHVDGKNlQrMuPrUglGOKZQhiPrTTEfWnmSmmSgBhGDRRnJzRTAfk0lFFIBcmjNJSZAoAdupC1JkU3PvU3YDt1PU561D+NKrc1QE0gHlsfauO1m8aNmQE4rrZG/dN9K4bXOZTxUslmPpimXVy/fFd5Fd+VCoJACiuF0eRI9RLMRgDvUfiHxVFbqYkcbjxxQBp+LvFaW8LW0LHew4xXG6FplxqtwJ7j5laqdn/psxnuJMgHo1bb67FZxeVbxjI9KQjuLSO20uBegIHY1S1LxdBApAzmuDXUdQ1KbYokA6V0GneFprv5pnP/AAKmMzb/AMT3l7kWrsM1HaaVq+ptmZsqfY139h4Rt7UqWCtj2robextoVCrEgx6CiwanCaX4FUAtLGCTXU2Ph+2tUx5VbYUDpxTuKAK8UCQjCjGKkzmn4FJigYyin4oKincAXHepFPNRDGaep560xk2RSGkz70Z96AGHg01jk044zTlAJ5NAEWPalUY7VPsWjatMCKlzUm1aNq0AN3cUzNSFRioyvNNDE30uaTFGKBC5ozSYoxRcBQc07FR520GQ+lK4DicdaaZFXkkCs7UtSSzhLuRxXl/iD4iCB3SI8jjg0nKxLZ6/9oj/AL6/nQJ0PQ185Q/Ei5Nxgl8Z9a77wx43TUJEhZgCe+aXOHMeoAhuaXFV7aQSRKwOQasg0yhMUhp/HrTSKBXG5ApN2TSMKRV+brQFyWilxRigLiUUuKMUBcSmE81JioyBmmh3Hg0u+mYpPxqguI5Fc34u/wCQJLXQvXPeLf8AkBy1LE2eeeEkDay+a9ktoIxAvHOK8h8H7RrD163DKfKXHpUIlMJLZBJnHFc14yijOl8DnJrqJHJQnFct4rP/ABLefehjuch4MX/TpwP71eqWyMIx9K8v8Fj/AE+4/wB6vVYWwg47UkK4/p1pd1I3zUbaY0Luo3Um2jbQMXdUdxGJ7aSM/wAQxT9tKooA8J8d2z6dfPsG1eazvB9wJNRHrkV33xI0sTW8s4FefeD4SmrFT2IpE9T3+xcfZIv90VbUiqNgD9lj/wB0VaXg0yifIpu9c0nWoGJDGmgLSsppTg1XRuKkU89aYD2XNRlB6VLnjrRxQBzmv6JFf2rKUyTXkE8uoeGNWYqWETP/AHT0r6A2LJxxXOeIfCttqUDEom4c5xSauKxV8PeIotStkIbn3NdLG2RkV4iz3HhbV0iZmMbGvVdC1yG/gUhl3GhMEb2CaQg1IpG3I70jEUxiKOKWmqTinUDHUUUUAIajdsCntUEretSIoXerRWv32A5qsPEFuwz5g4rjfiFdSW9unkNhiwH61Q8PaXqd7bO5OcjI4pCPS7XVYLgcSZz6VoIwY8dDXiWqXureHdXVZJMRAc4FeoeE9UGqWMcmc5HencEzoZB+6b6VyGsRAuTXaSR/uz9K5fV4wsbSHtQNnlOv6w+lSuyGvOb7V57q68wk/ez1r0bUdGk8S6rJZ2wy/WmD4Qatk4jH5VJBiaNczX8ILjao9K63TtNseHlmbd71BafDnxBaLsiCgfSrR8B+Jvb8qeo9To7CbTLM5V1P4VsJ4js16Oo+grgG8FeJVPVfypn/AAh3iX+8v5GmM9G/4Si2/wCeq1JH4ptc8yrXmv8Awh3iX+8v5Gl/4Q3xKf4l/I0XYXZ6Z/wlNp/z1Wj/AISm0/56rXmg8F+Jf7y/kaX/AIQvxL6r+Ro1A9K/4Sm0/wCeq0f8JTaf89VrzX/hC/Evqv5Gj/hC/Evqv5GjULs9K/4Sm0/56rSHxTadpVrzb/hC/Evqv5Gj/hDPEv8AeX8jQF2ej/8ACUWuf9atOXxTa/8APUV5v/whXiY/xL+Rpw8EeJ+zL+Rouwuz0oeKrT/nqtH/AAlVp/z1WvN/+EI8T/3l/I0f8IR4n/vL+Rouwuz0U+KbX/nqtA8VWo6SrXnJ8EeJ/wC8v5GgeB/E3qv5Gi7C7PR/+Ettf+eq0f8ACW2v/PVa85/4QXxP/eX8jR/wgvif+8v5Gi7C7PRv+Ettf+eq0f8ACW2v/PVa85/4QXxP/eX8jR/wgvif+8v5Gi7C7PRv+Estf+eq0n/CWWn/AD1Fedf8IN4m/vL+Ro/4QLxN6r+VNNhdnov/AAldp/z2Wj/hLLT/AJ7LXnf/AAgfib1H5Un/AAgPib+8Pyouwuz0U+LLT/nstH/CWWn/AD2WvOv+EC8Teo/Kj/hAfEv+z+VGoXZ6J/wldmf+Wy07/hKLP/nstebt4F8TJ3H5U3/hCfE3qPypXYXZoeOfFcRtZEjkHtzXiF3cNPcPIWY5PrXb+JfCur2sbPd8j2U1wciGORlPUHmpZLGVseHL97HVUm3kAeprHq9pMDXF6qAcmkI+gtH8ZQixiDSKDj1rSHjK3/56D8q4jSfDEklnGdnb0rRHhNyfuD8qu7LuzqR4zt/+eg/Kj/hMrf8AviubHhB/7g/Kl/4RF/7lF2B0TeMYD/GPypv/AAmNuOfMH5Vzx8JSf88xTf8AhEXbgIPyp3YWOk/4Ta3/AOeo/Kj/AITa3/56j8q5v/hCpv7go/4Qqb+4KNQ1Ok/4Ta3/AOeo/Kj/AITa3/56j8q5v/hCpv7go/4Qqb+4KNQ1Ok/4Ta3/AOeo/Kmnxpbf89B+Vc7/AMIVN/cFH/CFS/3B+VF2Gp0X/CaW/wDz0H5Un/CaW/8Az0H5Vzw8FTZ+4Kd/whMv/PMflTuwuzcbxnbn/loPyrG8ReKoLnSpIlkUk1F/whMv/PMVHJ4IkYYMeRS1DU5jw5rCWupPIzAA16ND4xtRGo8wce1cx/wghj5Cc0sfg+XnCD8qm7QrHVHxna7MeYPyrnvEnimCew2K4J+lQHwdLt/1Y/Kom8FPKNpjB/CgDN8La7FaXUzu4G5s13q+NLYKB5g/KuYj8BSR/diqX/hCZv7i/lQB0i+NrUdZR+VO/wCE4tf+eq/lXMjwHO3RFpR8P7j/AJ5rRdhdnS/8Jxa/89V/Kj/hOLT/AJ6r+Vc3/wAK+ue0a0h8AXGPuLRdhdnS/wDCcWn/AD1X8qa3jm1HSVa5k+Arj+4PypyeALhv4F/Ki7C7J/Enii2vtNePepJ9BXBaPqcNpqRlLgDNdfeeA7hIiSi/lXn93o00GomHHIIFIGev2fjS3WBB5g4HpUx8bW+f9YPyrkNL8GzTW6sUHK+laB8B3GMhF/KndhdnRL43t8/61aU+NbY/8tV/Kudi8A3DN/q1/Kpv+FfXH9xaabC7N1fGtsP+Wq/lTh43tR/y1X8qwP8AhX0/9wflTT8Prj+4v5UahdnQnx1bAf61fyph8eW3/PVfyrAPw8uT/Av5U0/Di6P/ACzX8qLsd2dFH47tt3+tX8qWbxzbMuPNXH0rnV+HdyhyY1px+HtweiL+VF2K7MfxjqdrqiGVWXeo4xWL4M8QT2t/HCzcD3rsj8O7llwVBFT6Z8PFs7tZXixRqFjvtLujdWsb+3arrK3pVXT4EtIVQAjFX94NUUiNRgYpaU9aSgY6iignFADGqldyiNC1XWI7VgeILwWtm7HjFSSeW+NL8Xt6kAkxtkGefevRtGms7TS4SJYwdgzzXkdtpj+IvEkyBmIByMGuu1LwjNZ6S7pNLlU/vGkJNmP41Y634gFvbjIYfeUZFej+BtKk03T4kkbJA5rzbwZcRrfrFcEGTJ+9yete1WOBGuBgUIEaMp+TrXNa4hNpIa6GQ5WsLWAWsnGKoo4HwauPGsmD/CP616yowpryjwz/AKN4xkduAQP616ujB4tw7ikhIhXhqlqvuOaeJDTQ0I6g/wANR7B6VYHPWl28UxlcIPSpVjXH3acTjtSebg8CgBCgH8NGwelOEm6lzQAzYPSjYPSn5ozQAzYPSjYPSn5oLUAOWNfSnqg9KjEnNPWTmgCUIvpRsX0pu8+lLvPpQA0ovpRtApGemF80AS/LR8tRc0c0AS/LR8tRc0c0APIXFRnPalwaKpDQ3Jo5p2RRmgQc0c0Zo3UCHRgE8ipDHHjoKi3EUhPFILmR4h0WDUrR4yq5NfPPivwbdafdySojsrHgAV9MsMiqN5pUF0uHjU/VahoTR8kLpl6zYFtKcH+7XqPw78FvJJFfXCkAdVYV6rb+FrKFmPkxnP8AsCtOPT4oV2xqqj0AxStYVrDbeyihjVFQYFWVt0/u1YjTCgVIAPSrLRAIU/u0nkx/3RVojAphpgVjCn9wU0QoDkKKnY0xWyaAG+WvpR5a+lT5ozQBB5a+lHlr6VPmjNAEHlr6UmwelWM0lCAhEa+lO2CpRT+1MCsY1z0qN4x6VbamNQwKXlA9VqzDbx7TlKdVmNvlPFSTYzrlYolJbCgdzVJbm03YEqZ+tU/GU/l6bKdxXA615N4fnudU8QPah32DB4bnvUvcR7ipjYZBBBHaonlhj4d1X6mm6fp/k2qAs2cd65LxuBawvJ5rKVHQGlcDrFvLfOFnQn0Bqyj5GQ2c1434W+0X94reY5U+rGvW7GIxwICTnFNDRdUk96kK5pkYqcLzQMr7Kcox0GKsbRTWWgDN1Qf6KxNeL3qb/ELj/bH869o1jC2LV4w3z+JHz/fH86TJZ6tpceyzix/dFbkCho8EVV021H2GE/7AqWW+gsMmaQKo6k0xl6ONVOSMVExwTWPP4v0o/KtyhbuKB4g09hk3K800wubGfeisj/hI9K3hDdLuNaVvdQ3ShonDA+lO4EwNSCm7afQMawyKaBipQAaHGBxQBH2phGetOpwXIpgRbacDSsAKiDEnikBLRS49/wBKXFAx2BTHHFPprdKAKzMVzXBeONQCWcqE13sykocV5v4x0S7v/MEbHB9qlksr+A7Fbe8N5j74rr9f1OG30udZSMsmABXD6RZahYRLHufgY6VNe2F/qDBGdip65FIDmfDlu134niuIgfLz/Wvd7NNluo71ynhfwumnxB2Ubq7JF2gAdqaBD+q1n6jFm0fNaHaqt8u61YUDPMrb5PEr49q9Qsn3Wqe4ryS/L2uvM+cDI/nXqukN5mnwN3K0kJFswDHSl8oVN2pBVopEapinbadRTGR7aTyhUtFAEIQDtTttOooAbto206igBu2kK0+kNAEeOacBzRS0hC0opKUUAJjNAXmnUo60AG2jbTqKAG7aNtOooAYV4phqY9KiPWqQxKXbRTqBDdtKFFLSjFAhNuBUZIAqRzx16Vh6trUOnqdxGccDNSDNR5VQEk8VTbU4Q+3PNcU2t3GpzYgLBT0qae1u0i37znFTcR2kdyJMYPFXolVlzXk9l4kktNUjgmc/Mcc16hYzrPbh1IIPcUIRZ2807FApao0E5pNtOooAjKZ7UgixUtIaAI9lGyn0UAM2UbKfRQAzbRtp9JTQDcU7miimAhGaQrmnUUCI9lPDbVPpQajlbbEx9qQjgviPfJHo1woPzY6VzXw/tbeAJfuPncdad46uGvL82an74NX9B0eeDSokBIxWfUk79dZt1TrXmPxA1UXeoC2iJIYV0k9hcR27OWPyjNcDcxtda9EW+bqKGB2vgPSPKtYpGXmvQVhxWX4ct1hsYxjGK3QBTQ0RKmKf0FPIprDigYwvTlO6ozTouKAMnX5Nlm4ryC3id/ETtjjcK9U8VyiKydz0FeeaMv2nVmZR3zmhiZ7BpoUafEM9EFcj43YCzlByBXU2T7LRQey1578R7/y9PmA+9j1oBnn/AIbsW1XxHJABlF5/nXqEXhGERrmPmuJ+HVxFbXhupF+Zl6mu0v8AxVEbyKKLjc2MCkhIwdW8MtFq8YiTAxXf+HLN7a0jVh0qSC2S72zMuSAOta8EYjUKKoaRLjig0/tTDTGKh5pZDmmAgUvXpQAzFOBwKDwKiZwBQBHNKACfauS1zxN/Z0qqrgE8Vo6zqS28LYI3YPevFPEOp3F5q0YDNt3VLYmz3fw3qTapponY55rZxXJ/D5Cvh5d3ByK62qRSCmsM06igCIrmonhjb7yA/hVnApNgosFii1lC/wDyzX8qVbOEEEIvHtV3YKXYKVgsQpGEGAABUgFO20YFFgsJjioLhd0RFWu1RMtFgseS+Lf9GvTIRj5v616PoLb9ItW9UrgPiJFiPcP71dn4Tu0uNIto1OSiYNLqStzpO1JTj0ptUihc0ZpKKYxc0ZpKKAEo5pwANLgUXC4zmjmn4FGBRcLjOaDT8Ck20BcjxS0/bS7BTsIYBS4p22jFFgExQKdRgUWAKKKKLAFFFFFgCo8U+kpbANpaDRRcAzRmm0UXAqapdra2kjk9ia8Z1TUrjWtWESFtu4g4+td38Q9Rays41BxvGP1rlvAmnC5lM0g535/WoerJe52HhjQxaWqNJyw9a37iBTEw2jGPSriQqgwBUVziOF2P901Vh2PEfHYFjq0UqHG054r034eX7ah4bjmbrkV5R8QWkvtcghh53HBr1P4c2Mun+G44ZBgjHBqUtSUtTshS5pBS4FaWNAzRmjAowKLAGaQ0uBRRYBtFOwKMCiwDaKdgUYFFgGGinECm0tgCiiii4BRRRQIAMmq2pypBaOxOMLU5kCA9q43xXqchiaGI5LDGKQM4ab/iY+OoFByuCP1Fev2VikVsq4HHtXkvhywuodcju51xg9a9bW8DQBlNJEoo6/cQWOlzb9qkoccV5f4ZhXU9R84cgSEfrXRePJLu/ijitueMGs3wFp0+m5WZcMWzzS6h1PT7KHyIQg7VbWmRYZQak6U7FDuKQjikop2Cw3YKb901JTHHNKwWOQ8cyZ0mVc81h+CbBWmLN3FXvGtypMkBPPpVnwfFsjVh6UupPU6kgRwN6AV5L45k+2XbWynJYHivT9Tu1trSQk/w15IrnVvGkSqcqQR+ooYM0tD8MyRaWsgDA47VFp3hy6fVPMfzMI+ec16xYWCQWSREdKctnHGxKjk+1Ow7EdhEY7dVPYVoIKiRAOgqVeKY7EnamNThQQDQBAxPapoxxQUUCm+aqd6AFl4FZV/dLawF3OAKvSXSY5PArjPEt6bsvaQHJPShsTOb1G7e+u5QrEqPSuamt4jqUS5BJauoi0DVI4nkEYwVNca8dxFrkSzDB31BLPcPCsSw6SFHtW7WN4ayNMX8K2a0SLQUUUUgCiiigAooooAKM0UzvQA/tTWpw5oK8UAed+PbffZMcZwSap/DHU/tFzcwFv8AV8YP0Fdb4m04Xdi6+xry7wXNJpniO7R1KhpMD9Kl6Ml7nuxOVFNpsbbokb1AP6U6qKCiiigAooooActLSLS0AFFFFABR3oo700AUUUVQBRRRQAUUUUAFFFFABRRRQA00lK1JUsAooopAIaSnUYGKAPNvijzFZjrg/wBal8CKiW2QADmrvjfSm1G3RlyfLGa5XwzqU2nTeTLEVG48mpe5L3PXwwzWJ4iv0tbFzuGdpHWqd34hjgti8bB3/u5riL641DxHc+V5MkaA/ep3Hcy9CtTrviBbh1JSOUjke9e428EdvGEiAA9K5nwp4ej0i2OQGZuc4rp1600gSJRRQKKsYUUUUAFFFFABRRRQAUUUUAIabTjTalgFFFFIAooooAesasORmsq+0y1ldS8WSK142A7VFKQT0FAGbb6Xaqn+qq99mjWLAXipYmAXoKkLjbQBlzadbucmLNLFp1shBWLGKvlh6UZHoKAGIAowOlOoooAKKKKACo5Dg89BUlUtSfyrSRs4wKAPNfF9ysniFoRzmuy8OW4isI3xjiuAmtmv/E6zFiwr1XTIRFYRrjGBSQkcp4raeWBo4twyuK4rwvpF3Za3FdzhmCn0r2SS3if70an6impZ24bIhUfhRYLE9tL50CvzzTsU5UCrgDAo70xiDrTxTcU4UAKDSg02loAcwyKoyfxelXl5NNeIMG460AcD4p1o6epiUncw4Iqp4VhfUNlzLlmNdJqvhNNSk3tL+dXtG0NNKhVFbOKVtRW1ItSmWzsX4/hNeQCZLnxEjMhwJc5Ne3ajp4vYfLJxXLf8IFEJ/NEoBzQ0DR0uksn2NRHjHHStCs3TdONhEELlsVpVaGFFFFSAUUUUAFFFFABUf8VSVH/FQBKtOI4piVKQMUAZ97HvhYY6ivG/EIl0nXrdwu0STAn869slHBrzXx9p3mvbzKMhG3E0mJnoOn3SXFnCUOfkX+VXBXG+CL77ZYH5vuYX8uK7Ee1NDQtFFFABRRx60UAOWlpop3FABRRxRxQAUd6OKSmgFoooqgCiiigAooo4oAKKOKOKACijijigBrU3JpW9qbzUsBwopKWkAUuKQU7igCpNaiZWUjINYdx4Ut5pN2MHOeldRkCjcKAOTTwhADyWrYs9GgtV+UD8q1Nwo3CgBixADFO2AUu8elIWBpoBRRSA0vFUAUUcUcUAFFHFHFABRRxRxQAUUcUcUAIabSk0nFSwCiiikAUUUUAOBApGANNpQKAFUAd6UlcYzTSKaetADvl9ad2qMA07FABRRRQAUUUcUAFYPii/jtNLn3HDFeK3ScDJ6DmvKPibqjC9ht4m4cY/SkxMZ4PeS+u45SBgmvV4F2wha87+HtkYbSNmHPvXpCjihAiNhQBzT2pKYx46U3FO7UnFABRRxRxQAUUcUcUAG7HeneZnvTduaYR1oAl3imls9KiBp6qWoAMmlyO9PKHFRHg0AO60tNXpTuKsAoooqACiiigAooooAKAtFIGwaAHhcdKU0wNTs0ARSVz/AIks1n02Y45VDiugfJqpeQedA6f3lxQB5l8Ob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proof	Plane Geometry	Math	Chinese
47	"如图, 在 $\triangle \mathrm{ABC}$ 中, $\angle \mathrm{C}=90^{\circ}, \angle \mathrm{ABC}$ 的平分线交 $\mathrm{AC}$ 于点 $\mathrm{E}$, 过点 $\mathrm{E}$ 作 $\mathrm{BE}$的垂线交 $A B$ 于点 $F, \odot O$ 是 $\triangle B E F$ 的外接圆, $\odot O$ 交 $B C$ 于点 $D$.

<image_1>
过点 $\mathrm{E}$ 作 $\mathrm{EH} \perp \mathrm{AB}$, 垂足为 $\mathrm{H}$, 求证: $\mathrm{CD}=\mathrm{HF}$."	['连结 DE.\n\n$\\because \\angle C B E=\\angle O B E, E C \\perp B C$ 于 $C, E H \\perp A B$ 于 $H$,\n\n$\\therefore \\mathrm{EC}=\\mathrm{EH}$.\n\n$\\because \\angle \\mathrm{CDE}+\\angle \\mathrm{BDE}=180^{\\circ}, \\angle \\mathrm{HFE}+\\angle \\mathrm{BDE}=180^{\\circ}$,\n\n$\\therefore \\angle \\mathrm{CDE}=\\angle \\mathrm{HFE}$.\n\n在 $\\triangle \\mathrm{CDE}$ 与 $\\triangle \\mathrm{HFE}$ 中,\n\n$$\n\\left\\{\\begin{array}{c}\n\\angle C D E=\\angle H F E \\\\\n\\angle C=\\angle E H F=90^{\\circ}, \\\\\nE C=E H\n\\end{array}\\right.\n$$\n\n$\\therefore \\triangle \\mathrm{CDE} \\cong \\triangle \\mathrm{HFE}$ (AAS),\n\n$\\therefore \\mathrm{CD}=\\mathrm{HF}$.']			['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ApDS0hoAaaSlNJQAU5WxTaPxoQmOLimb6DSYpgLy1J5eaUE04GgCNoNwIPOawdY8M2WoQt5kClscE10e4VG2KkR84+LvDU+i6gbi2BVV5worf8CeOzFKltdMTjH3q9P8QaPDqVq8bKuT3xXh2v+FrnSr5p4A+0HjFQ0S0fRFpex3UKSIRhhnip99eOeEvGrQ7LWZgCPl5r1azvUuog6sDkCmmNMxvFOqtaWbbImY89K8NvdSupNVaRrSVgr5+4a+jp7WO4TDqGz6is5vD1kTkwRZ/3RRYTPOdM+IIs7dUOmzZA/uNUerfEGa9tjFb2U8bnvsNek/8ACP2X/PCL/vgU5fD9lu/1Ef8A3zRYLGH4L0d4UXUHGHmGTkc126ZxTIIUhjWNVAA4GKkbAqkNDs0ZpoxS00UOU06mCnA1QCmm8UpptJjY7NGabRUiHZozTaKAFB5p9MHWn5oAKKM0ZoAKKM0ZoAUUtIvSloAKKKKCwooooAKKKKACiiigAooooAKKKKAClpKdigljCaaTTmpjUCFyKY0gUZJ4HNNLY5zWLrerR2Vq+WG7aR1oYFXxD4nt7CBkD4kPSvNksbvxXqqzSDdET1pkNre+JNXDbm8pX5z6V65oegw6ZbKgjww7ipepL1ItA0GHTLVFC8j1rfUAdKNvtS4qihaQ0tIaAEooooAKKKKACmnNOpKAG80vNLig4FAEbdKxdX1aGxhJZsH3qfWdYh023Z2IyPevKbnUrjX9WaKItsBHbik2Jsm1Ka71662R/MmccV1nhTwoLJUklTBrR8P+H47eFXkQbjzXWRRqiYAAoSBIasSouB0FG2pDTaYxKQ0tIaAGmlzSGkoAdmjNNooAdmjNNooAeKWmjpTuKACijijigAoo4o4oAKQ0tIaAGmkpTSUAB6U3cAadUZjzTQC7ge9KDTdhFIWx3qhDzQTiq7zYqCW7AzlgKlsC7vqNm4rDutcitslmB/GuS1fxqFfZDuyeOKlsVzuru+hgB8w1xXiXVtNnt3jVsvg9q5/7fqWpyfK0gU/7Nadl4SublmkmbII6GluG54/dXD22qtKhwN+RXrngXxak0KRTS/NnGM1w3jfRl06RcAZJ5wK5XT9TmsLhXRyAPSp2J2PrWCdJk3Kc5qUmvKvBfjIXKxxSyc98mvULeVZ4w4IIPNUmMcTQHIqTZmkMfFMYCSlDbqZjFOWmCH04U2nCqKAU4U0U4UxhSUtJSYBRRRUiCiiigAHWnUg607FACUUuKMUAJRS4oxQAq9KdTelLmgBaKKKCwooooAKKKKACiiigAooooAKKKKAAU+mU4mglkb0xzxSuwFVLq6SCFnbsKBGfrWqJp9szlhkV5Rc6heeINUeBFfYD1FaninU31O5e2tzkt2zXTeDvDS2cCXDJ87DnIqdydzW8K6HFp1qCygueeldMABUUKGMYqWqKCiiigApDS0hoASig0meaAFooFO2nFADajdttSE461WncAE0APMmASaxdZ8QQadC26RQQOM1X1nXIbKA5cA9q8wu21HxLqAWMbot2CQe1JsTZZvry98TX5jiV/KfuprvPDHhRLCJZZMFiOc1L4U8NR6TZoGX5/cV1qKB0AFCQJAkSqoAAAFG7FSY4qEjmmMdvNGabRQA6ikyaMmgBGFNpx5puDQAUUc0hoAWikyaM0AOFLTQaUUALRS4oxQAlFBpu73oAkpDSBqOtACGkpW4FQtKFFAEvFG5QOTVR76JM5NZt7rVtCpLSAAUXC5rTXKRoWLAVk3OrwRZy4rjdc8bW6wvHBLmQdAa4uS/13UZP3UeUPQ5ocibnpN74st4jgSLntzWBf+Irm5J+zxs+f7tZukeFLu8YPfKV+ma9B0fwvZ2oGASfep1YXZ5+lnqepSYkilQHvXQaf4GDYkllJOc816IthBGowo49qVY1B4xTsBi2OiW9ogHlqce1aBgVEbaoHFX1QYpsqgRtj0oGeE/EePzLyKMdWbFcPNorxzKhBG7mu/8AHYzrVqO3mVHrloIRHcqBhUHOPaoJOKhS50iQTJuwK9d8F+L47iJYpXAYDGCa4v7OuqaaNoBbFcglxcaNqRI4G6gR9UQTLKm5WyCKlPNeeeCPFsd7bLHLJ8/QAV6Cj7hx3q0MMUKMU8UGmNCU4U2nCqKAU4U0U4UxhSUtJSYBRRRUiCiiigBV60+mL1p9ABRRRQAUUUUAFFFFADqKKKCwooooAKKKKACiiigAooooAKKKKACkJwetLUUh5oJZDO+ATngc1wHi/X/KJgifDMMcV0+u6rHYWzEsMlcYrzazs5tc1ZZGztD9/SkyWang7RXu5Y7y5Xd65Fep2saxRhUGAKpaXpaWNqI1A/CtNVAHSmkNAaSlNJQAUUUUAFIaWlwCaAImJpKsGLgUwx4oAjyaQuRnmnkVBKQoJY4A5oAgmmYHrwK5HxP4nTS4Hy2WHpSeI/EsVmGijcM59DXL6do9x4mullmLCNuxqdyWzO083/ijUDtZhEGz8w7V6vo2g2unwLsiw2OSBUujeHoNLhUIihgOorZEYUU0u40u41Bg1KKbjFOpjJO1QHrTz0phoASilxRigBKKXFGKAEp1NpcmgBDTcU+l20AR4FG1fSpdvtS7fagCDFOH0qTaM9KcEoAYCacD607bgVBI+w0AS7QetIVUCqcl1t7VAb/14xQFy8eM1VlkZScNWXe6/aWiktOgI7E1yN947fe0dvb+bz1UUriudzLeiNctIox6muZ1bxnaaeCG+Yj+6c1yM02qa6+As0Ib0q5p/gGcuJLi6aQejUrtiuzNvPHf9oSmK1jkVicZ2mp7Lw/r+qjzDP8Auz2IrvrDwrY26KTBGW9cVuwWCRLiNQo9hVKI7HC6f4FgVg17Ernua6a28P2VuqiOLAHTFbX2bA60eVjvT5UBXitYoxwuKtRxqvQAUBSKcOKbAkzUR4PFOBpGHNIAUn1psxPlt9KctNnH7tvpSA8Q8cf8hy0/6611d5pS3fhx3wCwUdvauU8b/wDIdtP+utem6JEs2lIjDIKjioQkeU6C4tdU+yyg7R2q74u8GNNaLd26gZGTVzxHYfYNZedFwM9q67TpU1bSREcErHigR8/WOpXujXoCMQVY9K908DeLV1O1igmb98cZJrzrxRo4sNTB8nK5zmq+nXcmlzjUYCSv9wUr2EfRIYkZ9adXIeGPFsOqQqjlUcdQTXWI4dcg5zzWiZSH04U2nCqKAU4U0U4UxhSUtJSYBRRRUiCiiigBV60+mL1p9ABRRRQAUUUUAFFFFADqKKKCwooooAKKKKACiiigAooooAKKKKACqtzKsSl26VZPSuR8basLLSmCNhwaGyZHCeK9Ykv78W8TbgrAda7LwbpfkW2915ODXHeGdFk1G/a5lG4Mc163Y2y20KKo7UkSkaCdKdjiowwAqUHIpjI2pKc/Wm0AFFFFABQvBoooAkeQACq8twAetJMxrOvJhGNxIAFAFt71EUljxXGeJfFsUMZhgkBdhtxWf4l8RhIGht2IkrnNB0O81W+M1x86lsipvcncl0nw/c61eC4uA2M9q9Y0bSI9OgVFUceop2kabFZQKoXBxzitXIqkhpAQKaaXNNJoGFIaM0lAC5NJRSigBKKdgUYFADaKdgUYFADDSc04iigBKXdTCaTIFAE24UbxUBYUhcDvQBPvxTTOFqCSVQDkgVj32tW9sDubP40XC5ttdjFZ93qsEGTK4FcZqfi+II3kbs9q4+51LVdUmKxudp/2TSuTc9A1HxXZR/KkwLVy1/4o1OVitpGGB96m0TwjPeHfOoY12dl4UtrYKXj5+lLVhY8sGk6nrFyWuVdc/wB0muw0LwXHbfO4Jz1zXax6ZBE2USr0UYQYp2GkZ1tpcUIG1B+VXRDt7D8qthaXaKYysBjtViFuOaUxZFRn5elNATMQaZxUYY5pcmmIWkIpaBzQAgFBp2KRhzUgIKbOP3TfQ09RzTLn/UP/ALpoA8T8YwvPrtqUGcS816foClbFAR2Fec3cgn15lbnbLXp+lLi1UewqEJHOeNrDzLFpFHPNYHgrUvJupbdz3wK9A1m0F1ZlOvFePys+k65nOAZBTYM7zxdoa31k8ipklR0FeeaLZoLwafc5VR3717FYzLqGmgk5+UZrzTxjZPpVy9/Fxg9aTEzKm0680LUPtFspMbNnk9q9Q8N6/He2yqz/ADAYOKzdLNrrugxqRmUIc59a46drrw/qQUHEbNk8dqExJntSyK2CDmplGa5Pw5rSX0KHdyfWurjYEA1XMVcdtPpRgipQ3FNbmqTKQ3FJin0YFDGMxRin4FGBSEMxRin4FGBQA0DmnUDrTsCgBtFOwKMCgBtFOwKMCgBtLS4FGKqwBRTttIRSKuJRRQKLCuFFLikosFwooopDuFFFFAXDijNNNMZiKVxXC4mWCBpGPC968h8RXjazrz2kZyh/xrs/GesrZaPPGGxIRkCuG8CWU2oXiX0i53d6TJbu7HpXhrS1s7KP5RuxW6wA6UyBdkKrjGKkPPWqKsR5NTxn1qLYKcOOlA7Ej9aZTS2TT+KAsJx60cetMYHPHSmb8daBWJqM1H5o9ar3N4sEZYsAKAJLqRIoyzMMDnmvNvFfifMptbb5ie60zxP4waVjbWzKzH5cA1U8J+FptQuRe3isGU9DU7kbi+HfD0+oOs9zuIbqGr0rT9LhtIlVY1BFXLbT4rePCqv4CrWzAppWKSIlGKUtQ3B4pAN1MAyaSnbMUmKAAUtGKKACnCm0u4UALRmkyD0oxRcBaKSlyM0AB96acetJLIqjrVN7oLyTRcLlhj71G7Ack1k3uuQWyks+MetcvqPjWJcpE4LdAMUriudrNeRxKSZAMVzd/wCMLe0mMeVJ+tcdNq2q6gxSCEsG9DRbeCLy/mE94JYye2TS5mLmuT6j4zmuHKQI/J7VTitNS1RwWMqg12uleArO2CyFyzehrqbewit1CqicdPlppDscLpvgh2jDSyZ9jXR2Xhq3tT/qlJ9cV0aKF6AflTqYWK1vAkIwqgVa+UjFRkHtQqnNAxfKFJtxUtIRQBHmnbvWgr6Um00APD00rvpuPapEOB700AzYRSbTUzEGm1QWI8GlHFOpMUgsLmmnrS4pjNg1Ihw60y5/1D/7pp6jjdUc53QSD/ZNIDxd/wDkYpf+uteraV/x7L9BXlL8eIpf+uteq6V/x7L9BUolGg0e9SMV5b400UpMk6jHzZ6V6ulYHiizFzaHAyQpNX0LexkeArv7RYSIx5Ukc/WrHjXRft+kyIFyx9q5bwte/wBmaqtoxxvc16rdqssBGMg+tQSeGeCNUksNduLOUnap2gHpXo2saDFqtn5qgbtgwa878T6TJo2si9jUgSygk/jXqnh7UI7ywjAIO1AKRJ5lDLc+HtSET7ti16ZomrpeW6ncMkVm+LPDy30UkqL857CuK07VJdDvPJl+VQQBk0AeyrLkDuKkVsisPSNTivoQyPuOOcVtRkFa0RaHmlHSkNKOlMoKKKKQgooooAB1p1NHWnUAFFFFABRRRQAUUUVaAfSNS0jVKAbSikpRTAKDRQaAEoooqQCl7UlL2oAYagmYKpYngc1Oay9ZmEOnzPnG1DUgeXeOb433iOCzQ5Rxg13vhHSE07T40C4x7V5/4dRda1v7U6hjHIRk9ua9jt41jiwBimhR3uS0UUUywooooAKUHmmk4NIDzzQBOFLCqlyQnU1Y3jBGcVl6lcLFGzswwvNAmyKa9jhQsx6VwniTxR5uba3Y78+tUfEXilpZmtbdDubPK1e8KeGnuJFvbtd4bswqdyL3HeFPBy3ErXeoR7gx3Ia9IsrWO1QJGMAUkEKRRKqqAAMYqygqiifNITSCigCJ+tLGcGhutMyQeKAJmOaZSCnqKAGjPpShDUgAxzTximBF5TYqLy2Bq3lQOtQO656ikwGKhXrTt4AqvNMqgneBWTdapDboxaZOO2aQrmu9wgNVJr+KI/McfjXA6n8RIbGUqsBl91rmrrXtR8SS7bWKe3DdDtpXFc9D1LxZZ23DNnFcnqnjdbg7LRmzVXSfAOqTS+bd3bOrHOGxXf2HhLTrZF8y2Rmx1xS1DVnmiaf4h1s5Q5RvUGui0bwCA2+/iyc54r0CKwt7f/Uxqn0FWVT15p2HYyrHQrKzZTHHgj1FbewFcHpSKAO1Shlx0pjIwoUYpaCeeKKYBRRRQAUUUUAOooooAKKKKACm96dTe9NDQDrS0g60tNjYUUUUhCGoXHzVK1RsMmgCVf8AUmoJP9VJ/un+VTjiLFV5f9TJ/un+VSSeMSE/8JJKP+mter6SP9GX6CvJZG/4qWT/AK6165pI/wBEX6CpRKNNKgvYllgkB54qccCkK71Iq+hfQ8S8SpJpWvw3KcKhzXqHhnVv7W0WO4Jzn1rm/HukedaySKOQvHFZHw51J4jFp8jEEY4NQyGdR4x0kahZbgM+Xlq57wJqqLJLAzfdbaPzr026thLYSjAOUOK8EufO0fxTGqEokkpPFIR7w22VPauD8WeFlu0MsSfNnNdjo0wubJSWBJrQktUdDuAOfWmho8X0HXJdFvBbXD4BbbXrml38V7biRD1rkPFfhBblWmtwEZeciuQ8P6zeeH9WjtLt3ZO5bpTWg9j28nNKOlUtPvor62WaNgQ3pV0dKssKKKKQgooooAB1p1NHWnUAFFFFABRRRQAUUUVaAfSNS0jVKAbSikpRTAKDRQaAEoooqQCl7UlL2oAjauN8a3fl2RizguMCuwlYIpYntXlXj3VBJf2UETZJkAYfjUsTLHw00g21tM7gktIzc+5r08dKwfD9oLWzj4xuUE/lmt8VSKiFFFFAwooooAjkJ3CkYkDNK65+Y9qy7/VrewjJmbGKBXsM1HUVs4zI7YrzrWvFVxqE32e3XKsduQad4g1yXWZDBaNkE/pWl4X8JMH86ePknNTuRuHhjwl5rrc3Gd2c4avRYLOO3QIqgAdMCn20CW8QVRUpbNUi0hNvtSqMU4DijGKBC5NHam0uTQAhpNmafgUZCigBAmKUnb2qNpwM1Tlvgp5PSgC7JMVFQPctjise71+2twfMbFc7e+MrckpDIS3ak2K510+orApMjbRjqa5fU/GVrbMVjnUt6Vyt7f61qrbUwUJ/SksPAtxdyCS5iGTSu2K7Y6/8Y6ldnbbwbwf7tV4tH1HW2Bm82MexNd/pfg+3tEUNH0roYdPihACDj6UajscFpXw9iRA0rsxP97JrtNN0C0solQRRkj/ZrVUBR0p24CnYLDPJjUYCgfhSeUKczigMKYxnl+9IUIqUc0pFAERJxTPMPpUrDioytADl5p1IKWgAooooAKKKKAHUUUUAFFFFABTe9OpvemhoB1paQdaWmxsKKKKQhrU1RmnPREOtACtwuKrTHEEn+6asydaqXP8Ax7yf7p/lUsk8VkY/8JNJ/wBdq9h0gf6In+6K8ZiSWfxRLgcCavatKQraqD6CpQkXTRk0pppq+hXQztWtFu7Z1buK8njibRfGXmDhB27da9llGUxXlnj61e0El2gwc9alks9Psbz7Rp6N2Za8r+ImmsmpQ3EacLySK7XwlqkN3o9uitmQLyKTxZYLdaVO5GWC1JJmeAr8z2UYJJOa77dxXkXgS5a0nSFjg5r1qNw8QYd6ECIplEq7SB+VcD4v8Mi5hkniBD9iK9AxzUVxAs0RUjINWWeS+F/FsmkXv9n3JAVMAEnmvXLS8S6t45FYEMM15V4v8GsJDeW0fz5zUnhbxY9o32S7fBX5RSvYV7HrgGRxTtgqlYX0V1EHQ5q7u44qihCopppS3pSDmgBB1p9Jt70tABRRRQAUUUUAFFFFWgHZoY5ptFSAUopKUUwCg0UGgBKKKQctUgKeKAeKV+lNA4oAytavFt7N29Aa8VsGk13xPKGywimyP0r0rxtcmKxYZx1Fcn8MbATateyuP4sgn6Cpe5L3PVrWDZBEBxhAP0qyTt4qZQAoAFV5+HNUihwbNKTio0NK5FAx24Ux5gvJ4qPdjk1z+uazFZREmQA+hoYmzT1PWYLOzldmXIGQM15DqWq3ev6iYrdmCE1LqN5e65ceXArlDwSprrvCPhdbKNHlGX96ncncd4Z8KeRGks6Amu6ggjgjAVQPpU0MKqigAACnMtUUROaaDzTmWkCjPNA7kwPApCcDrQQABzVeeXYO9AibIHem+YAazZL4KCWbFZN14hghzmVRj1ouFzpjdIvpVa41GGNSS6j8a4W98V+apWLk9sVz89xqt85EcUhB9DSuK52up+Lbe2Ugcn2NcfdeI7m/dhBvXJ9KdYeEru/kBuDIgPrXa6V4Qt7EA7g/rmlqxanD22lalqCfvHY59RW/p3gl8q8oU/Wu8jtoUGBEo+gqwqKvQAYqrDsZ1hoVrboMxjNaiW8Mf3VxS54pN1AwYCm5pSSabigAFLs3GjFFADZFwKYM1K3NN20AOWncU3pRQAh6VGetS03ZQA0dKdRtpcUAJRS4oxQAlFLikNADuKOKbRQA7ijim0UAO4ptFFNDQDrS0nelpsbCiigUhDX6U+DGDSFQarvKYzgUATy43VVuh/o8n+6amR/MGajucfZ5P900mSeR6PGp8RXWR/y1r1uyGIVx6CvK9GQf8JDd/wDXWvU7T/Uj6VKEiyaaadSEVXQoYRmub8VaUNR0549oya6U1DMgdMEZqWSzxXwdq0lj4hubN2IVGwM163cKLvTGB5DKK8f8SaVLomvG8VSBLIP516roN2l5p0YyD8ozUknn4xp/iRYhxj/GvV9Ok8yxib2ry7xbbtb+ImmUcYru/DF2ZrGJT1ApoaOg200ipB0pp61aLRVubdJ4irgEY715p4k8KtFKbm3Xbg54r1UpmqVzZpNG6MAc+tDBnmnhXxHLbyrbTMc5716da3QnjDBgc+leYeJ9BksLg3NsCSvOFqx4V8UOkiW9x8pHXNTsTsepLzipQoAqpZ3CTxK6NnPNWy3FUUBxSU3dlsU6gAooooAKKKKACiiirQBRRRUgFKKSlFMAoNFBoASmg/NTqjH3qkCYjOKXZhKQHimzSrHCSe1AHmfxDulSEoT/ABVo+AdN+yQG4xxIM1zPjq4W9umhTruH869H8PQCLRLUdD5fNT1JW5so+WFRzffNIpw3tSykFqpFDFyKbK2B9Kf0XJ7c1zHiLxJBp0LLnLH0NArkmu67FptszM+CK8rmu7/xHqbRx/NHuFXFN54pv9qE+U/qK9J0HwtbabboxjHm45NTuLci8O+G47C2VtmWbk8V08cG1umBTo/kGD0qQuoFUMeGCjBqN51HeoXnXJ5qpLcIOpGBQO5baYHvTWuAozWVLqMESljIg+prjtf8dQ2uY41ZiO6jNJsTZ3E+rRx5y2MVz2p+Lre3ViZOlefQ3Ws65cEQPIiuf4lPSuq0zwFfyYN9IsgPUUrtiu2YuoeLbjUAUsW3sfem2Ph7XdSYPNF8j+hr0ex8FaZa4It1DDuCK3oLdLdAqAAD0p2HY4/R/AltGA06kN3zXU22i2tmP3a5+orRUjvTjz70xlXylHQAfhRsIqztHpTcCgCID1p4pxApKAF4ptFFNAFFFFOwBRRRRYAoooosAUUUUWAKKKKLAJxS5ptFSA7NGabRQA7NJSUUAFFFFABRRRQAUlLSd6aGhR1paQdaWmxsM0mcUh4pjHFIRIWBqrNGS2RUiHJ61KwzQBFCCFwajuztt3J/umpwOcVW1LItH+hpEnl+inOv3Z/6aV6jaH90PpXlmhA/25df9dK9Ptd3lDr0qUJF0HNLzTU9xT+PWqKGGmEU8mkNSyWcZ490xr+xjKoCYzuNY/gK/LCWJz91iOvvXf6jCs1jMGHPlkV4jZ6hJoniLyM7RLIcVJJ2/jWH5pJx2q94Mn3qq+gqv4lIuvDrzA8+v4VX8CFvOIPpQB6MG4pMjNNXlaQ5DVaLRMnJpjjIIp8XU80n8VMZm3lgt0hVx1rzHxP4dn064a6tkPWvZwikdqz9UsIryApImaGgZwfhDxMD/o9w/wAyjbjNehJOsiBlPGK8d8RaRNo9159uCq7snFdF4Q8ULeRiGZsPnHNSnYlM9CTl81LUUJVk3DBzUowaooKKKKACikz6mjNAC0UmfejNUAtFFHFIApRSUCmAtBopDQAUz+Kn8U3HOakB2QBVO9fFu3NWX4WsrUJCIGFAHmeqxmXWmAXdyP516ppY26XbjGMJivOocPrzbh6V6TajFpGB6VKJRIxwCelVnlHJLYqW7YR2zOT0FcFrviZLeFo4HBk9DRcbZq+IPFkGnQFI3V3YbcA1wVtpt34mvxO7usYPeoLHRLvXr/zpw23dng16rpGkx6baKqryB6ULUQ3Q9Dg0uBVAUsO9bxIC9az3mCZJNZ95r9rbKTJLincZsTSYYfNioJLqOFcySBQPWuE1LxohbFpIGx71iXWsa1qqGNIgUPcGlcVzttT8U2dmpKTxu3pmuPvfHN9dM0UFg7A8blFGm+BpL5xJc+YCfc13Ol+E7SxjHygn3WjUDzeHRNW12ZXaWeAHtmux0X4frb7XuZfOP+1XaQ2cUX3VX8quKQBinYEinbaTZWyqqW8akdwK0FAAAxUXOeKkUmmUTVGRTgTS0AV2U/3qfCpB5NSbV9aQ7U70AOJpKZvFG/NAD6Q0gaigBKKXHFJTQBRRRVAFFFFABRRRQAUUUUAFFFFADaKQ9aMmpAWikyaMmkAtFJk0ZNAC0UUUAFFFFABTe9OpnemhoeOtOxTV6innpTY2ROcGq086RIWdtqjvU8lc94skMeiykZB9qTEyaTxDpsDEG7iyO2aaPGGm/wDPzF+deGx6fcapqsih3x2wxrfj8BXLoCPN6epqOZsjmZ6kPF+m+Z/x8xf99VBqXizTmtWC3MRJHrXmn/CCXIcDMn5ms/WfCNxYWwlBk6+pouwuzpvDN0kmsXLqAQz5FeqWtyvlD5O1ePeA4P37hicg17BaRR+WMk9KECLQbzOgxS+XSfKv3TS7z6UFB5dHl0bz6Ubz6UCGvAGRlJ4NeP8AxE0U2mqQXUS5C8nFexbz6Vz3i3TlvtKncgFlXikJo4Oy1H7f4YKN3NanhOQQXJGK5OzmXT9NMDHBB6GrvhvV2OpFARjIxQI9mtZAY8le1RzP8/SmWTZtlJ7rUjpubNMoakmDxUwHQ96jWLFP3GmhjxxSMc0mTTG3UwM7VdNivLdkZFztwOK8j1rRrrQ7/wC0228qpyQte0tvwRisfVNNF5Cyso5pNEswPB3i37ekdvONkh7Gu7jfdgjkGvFNR0+48P6sbuNSFHrXpXhvX7XULSNRKDKB8wHY0J2BHVDGKNlQrMuPrUglGOKZQhiPrTTEfWnmSmmSgBhGDRRnJzRTAfk0lFFIBcmjNJSZAoAdupC1JkU3PvU3YDt1PU561D+NKrc1QE0gHlsfauO1m8aNmQE4rrZG/dN9K4bXOZTxUslmPpimXVy/fFd5Fd+VCoJACiuF0eRI9RLMRgDvUfiHxVFbqYkcbjxxQBp+LvFaW8LW0LHew4xXG6FplxqtwJ7j5laqdn/psxnuJMgHo1bb67FZxeVbxjI9KQjuLSO20uBegIHY1S1LxdBApAzmuDXUdQ1KbYokA6V0GneFprv5pnP/AAKmMzb/AMT3l7kWrsM1HaaVq+ptmZsqfY139h4Rt7UqWCtj2robextoVCrEgx6CiwanCaX4FUAtLGCTXU2Ph+2tUx5VbYUDpxTuKAK8UCQjCjGKkzmn4FJigYyin4oKincAXHepFPNRDGaep560xk2RSGkz70Z96AGHg01jk044zTlAJ5NAEWPalUY7VPsWjatMCKlzUm1aNq0AN3cUzNSFRioyvNNDE30uaTFGKBC5ozSYoxRcBQc07FR520GQ+lK4DicdaaZFXkkCs7UtSSzhLuRxXl/iD4iCB3SI8jjg0nKxLZ6/9oj/AL6/nQJ0PQ185Q/Ei5Nxgl8Z9a77wx43TUJEhZgCe+aXOHMeoAhuaXFV7aQSRKwOQasg0yhMUhp/HrTSKBXG5ApN2TSMKRV+brQFyWilxRigLiUUuKMUBcSmE81JioyBmmh3Hg0u+mYpPxqguI5Fc34u/wCQJLXQvXPeLf8AkBy1LE2eeeEkDay+a9ktoIxAvHOK8h8H7RrD163DKfKXHpUIlMJLZBJnHFc14yijOl8DnJrqJHJQnFct4rP/ABLefehjuch4MX/TpwP71eqWyMIx9K8v8Fj/AE+4/wB6vVYWwg47UkK4/p1pd1I3zUbaY0Luo3Um2jbQMXdUdxGJ7aSM/wAQxT9tKooA8J8d2z6dfPsG1eazvB9wJNRHrkV33xI0sTW8s4FefeD4SmrFT2IpE9T3+xcfZIv90VbUiqNgD9lj/wB0VaXg0yifIpu9c0nWoGJDGmgLSsppTg1XRuKkU89aYD2XNRlB6VLnjrRxQBzmv6JFf2rKUyTXkE8uoeGNWYqWETP/AHT0r6A2LJxxXOeIfCttqUDEom4c5xSauKxV8PeIotStkIbn3NdLG2RkV4iz3HhbV0iZmMbGvVdC1yG/gUhl3GhMEb2CaQg1IpG3I70jEUxiKOKWmqTinUDHUUUUAIajdsCntUEretSIoXerRWv32A5qsPEFuwz5g4rjfiFdSW9unkNhiwH61Q8PaXqd7bO5OcjI4pCPS7XVYLgcSZz6VoIwY8dDXiWqXureHdXVZJMRAc4FeoeE9UGqWMcmc5HencEzoZB+6b6VyGsRAuTXaSR/uz9K5fV4wsbSHtQNnlOv6w+lSuyGvOb7V57q68wk/ez1r0bUdGk8S6rJZ2wy/WmD4Qatk4jH5VJBiaNczX8ILjao9K63TtNseHlmbd71BafDnxBaLsiCgfSrR8B+Jvb8qeo9To7CbTLM5V1P4VsJ4js16Oo+grgG8FeJVPVfypn/AAh3iX+8v5GmM9G/4Si2/wCeq1JH4ptc8yrXmv8Awh3iX+8v5Gl/4Q3xKf4l/I0XYXZ6Z/wlNp/z1Wj/AISm0/56rXmg8F+Jf7y/kaX/AIQvxL6r+Ro1A9K/4Sm0/wCeq0f8JTaf89VrzX/hC/Evqv5Gj/hC/Evqv5GjULs9K/4Sm0/56rSHxTadpVrzb/hC/Evqv5Gj/hDPEv8AeX8jQF2ej/8ACUWuf9atOXxTa/8APUV5v/whXiY/xL+Rpw8EeJ+zL+Rouwuz0oeKrT/nqtH/AAlVp/z1WvN/+EI8T/3l/I0f8IR4n/vL+Rouwuz0U+KbX/nqtA8VWo6SrXnJ8EeJ/wC8v5GgeB/E3qv5Gi7C7PR/+Ettf+eq0f8ACW2v/PVa85/4QXxP/eX8jR/wgvif+8v5Gi7C7PRv+Ettf+eq0f8ACW2v/PVa85/4QXxP/eX8jR/wgvif+8v5Gi7C7PRv+Estf+eq0n/CWWn/AD1Fedf8IN4m/vL+Ro/4QLxN6r+VNNhdnov/AAldp/z2Wj/hLLT/AJ7LXnf/AAgfib1H5Un/AAgPib+8Pyouwuz0U+LLT/nstH/CWWn/AD2WvOv+EC8Teo/Kj/hAfEv+z+VGoXZ6J/wldmf+Wy07/hKLP/nstebt4F8TJ3H5U3/hCfE3qPypXYXZoeOfFcRtZEjkHtzXiF3cNPcPIWY5PrXb+JfCur2sbPd8j2U1wciGORlPUHmpZLGVseHL97HVUm3kAeprHq9pMDXF6qAcmkI+gtH8ZQixiDSKDj1rSHjK3/56D8q4jSfDEklnGdnb0rRHhNyfuD8qu7LuzqR4zt/+eg/Kj/hMrf8AviubHhB/7g/Kl/4RF/7lF2B0TeMYD/GPypv/AAmNuOfMH5Vzx8JSf88xTf8AhEXbgIPyp3YWOk/4Ta3/AOeo/Kj/AITa3/56j8q5v/hCpv7go/4Qqb+4KNQ1Ok/4Ta3/AOeo/Kj/AITa3/56j8q5v/hCpv7go/4Qqb+4KNQ1Ok/4Ta3/AOeo/Kmnxpbf89B+Vc7/AMIVN/cFH/CFS/3B+VF2Gp0X/CaW/wDz0H5Un/CaW/8Az0H5Vzw8FTZ+4Kd/whMv/PMflTuwuzcbxnbn/loPyrG8ReKoLnSpIlkUk1F/whMv/PMVHJ4IkYYMeRS1DU5jw5rCWupPIzAA16ND4xtRGo8wce1cx/wghj5Cc0sfg+XnCD8qm7QrHVHxna7MeYPyrnvEnimCew2K4J+lQHwdLt/1Y/Kom8FPKNpjB/CgDN8La7FaXUzu4G5s13q+NLYKB5g/KuYj8BSR/diqX/hCZv7i/lQB0i+NrUdZR+VO/wCE4tf+eq/lXMjwHO3RFpR8P7j/AJ5rRdhdnS/8Jxa/89V/Kj/hOLT/AJ6r+Vc3/wAK+ue0a0h8AXGPuLRdhdnS/wDCcWn/AD1X8qa3jm1HSVa5k+Arj+4PypyeALhv4F/Ki7C7J/Enii2vtNePepJ9BXBaPqcNpqRlLgDNdfeeA7hIiSi/lXn93o00GomHHIIFIGev2fjS3WBB5g4HpUx8bW+f9YPyrkNL8GzTW6sUHK+laB8B3GMhF/KndhdnRL43t8/61aU+NbY/8tV/Kudi8A3DN/q1/Kpv+FfXH9xaabC7N1fGtsP+Wq/lTh43tR/y1X8qwP8AhX0/9wflTT8Prj+4v5UahdnQnx1bAf61fyph8eW3/PVfyrAPw8uT/Av5U0/Di6P/ACzX8qLsd2dFH47tt3+tX8qWbxzbMuPNXH0rnV+HdyhyY1px+HtweiL+VF2K7MfxjqdrqiGVWXeo4xWL4M8QT2t/HCzcD3rsj8O7llwVBFT6Z8PFs7tZXixRqFjvtLujdWsb+3arrK3pVXT4EtIVQAjFX94NUUiNRgYpaU9aSgY6iignFADGqldyiNC1XWI7VgeILwWtm7HjFSSeW+NL8Xt6kAkxtkGefevRtGms7TS4SJYwdgzzXkdtpj+IvEkyBmIByMGuu1LwjNZ6S7pNLlU/vGkJNmP41Y634gFvbjIYfeUZFej+BtKk03T4kkbJA5rzbwZcRrfrFcEGTJ+9yete1WOBGuBgUIEaMp+TrXNa4hNpIa6GQ5WsLWAWsnGKoo4HwauPGsmD/CP616yowpryjwz/AKN4xkduAQP616ujB4tw7ikhIhXhqlqvuOaeJDTQ0I6g/wANR7B6VYHPWl28UxlcIPSpVjXH3acTjtSebg8CgBCgH8NGwelOEm6lzQAzYPSjYPSn5ozQAzYPSjYPSn5oLUAOWNfSnqg9KjEnNPWTmgCUIvpRsX0pu8+lLvPpQA0ovpRtApGemF80AS/LR8tRc0c0AS/LR8tRc0c0APIXFRnPalwaKpDQ3Jo5p2RRmgQc0c0Zo3UCHRgE8ipDHHjoKi3EUhPFILmR4h0WDUrR4yq5NfPPivwbdafdySojsrHgAV9MsMiqN5pUF0uHjU/VahoTR8kLpl6zYFtKcH+7XqPw78FvJJFfXCkAdVYV6rb+FrKFmPkxnP8AsCtOPT4oV2xqqj0AxStYVrDbeyihjVFQYFWVt0/u1YjTCgVIAPSrLRAIU/u0nkx/3RVojAphpgVjCn9wU0QoDkKKnY0xWyaAG+WvpR5a+lT5ozQBB5a+lHlr6VPmjNAEHlr6UmwelWM0lCAhEa+lO2CpRT+1MCsY1z0qN4x6VbamNQwKXlA9VqzDbx7TlKdVmNvlPFSTYzrlYolJbCgdzVJbm03YEqZ+tU/GU/l6bKdxXA615N4fnudU8QPah32DB4bnvUvcR7ipjYZBBBHaonlhj4d1X6mm6fp/k2qAs2cd65LxuBawvJ5rKVHQGlcDrFvLfOFnQn0Bqyj5GQ2c1434W+0X94reY5U+rGvW7GIxwICTnFNDRdUk96kK5pkYqcLzQMr7Kcox0GKsbRTWWgDN1Qf6KxNeL3qb/ELj/bH869o1jC2LV4w3z+JHz/fH86TJZ6tpceyzix/dFbkCho8EVV021H2GE/7AqWW+gsMmaQKo6k0xl6ONVOSMVExwTWPP4v0o/KtyhbuKB4g09hk3K800wubGfeisj/hI9K3hDdLuNaVvdQ3ShonDA+lO4EwNSCm7afQMawyKaBipQAaHGBxQBH2phGetOpwXIpgRbacDSsAKiDEnikBLRS49/wBKXFAx2BTHHFPprdKAKzMVzXBeONQCWcqE13sykocV5v4x0S7v/MEbHB9qlksr+A7Fbe8N5j74rr9f1OG30udZSMsmABXD6RZahYRLHufgY6VNe2F/qDBGdip65FIDmfDlu134niuIgfLz/Wvd7NNluo71ynhfwumnxB2Ubq7JF2gAdqaBD+q1n6jFm0fNaHaqt8u61YUDPMrb5PEr49q9Qsn3Wqe4ryS/L2uvM+cDI/nXqukN5mnwN3K0kJFswDHSl8oVN2pBVopEapinbadRTGR7aTyhUtFAEIQDtTttOooAbto206igBu2kK0+kNAEeOacBzRS0hC0opKUUAJjNAXmnUo60AG2jbTqKAG7aNtOooAYV4phqY9KiPWqQxKXbRTqBDdtKFFLSjFAhNuBUZIAqRzx16Vh6trUOnqdxGccDNSDNR5VQEk8VTbU4Q+3PNcU2t3GpzYgLBT0qae1u0i37znFTcR2kdyJMYPFXolVlzXk9l4kktNUjgmc/Mcc16hYzrPbh1IIPcUIRZ2807FApao0E5pNtOooAjKZ7UgixUtIaAI9lGyn0UAM2UbKfRQAzbRtp9JTQDcU7miimAhGaQrmnUUCI9lPDbVPpQajlbbEx9qQjgviPfJHo1woPzY6VzXw/tbeAJfuPncdad46uGvL82an74NX9B0eeDSokBIxWfUk79dZt1TrXmPxA1UXeoC2iJIYV0k9hcR27OWPyjNcDcxtda9EW+bqKGB2vgPSPKtYpGXmvQVhxWX4ct1hsYxjGK3QBTQ0RKmKf0FPIprDigYwvTlO6ozTouKAMnX5Nlm4ryC3id/ETtjjcK9U8VyiKydz0FeeaMv2nVmZR3zmhiZ7BpoUafEM9EFcj43YCzlByBXU2T7LRQey1578R7/y9PmA+9j1oBnn/AIbsW1XxHJABlF5/nXqEXhGERrmPmuJ+HVxFbXhupF+Zl6mu0v8AxVEbyKKLjc2MCkhIwdW8MtFq8YiTAxXf+HLN7a0jVh0qSC2S72zMuSAOta8EYjUKKoaRLjig0/tTDTGKh5pZDmmAgUvXpQAzFOBwKDwKiZwBQBHNKACfauS1zxN/Z0qqrgE8Vo6zqS28LYI3YPevFPEOp3F5q0YDNt3VLYmz3fw3qTapponY55rZxXJ/D5Cvh5d3ByK62qRSCmsM06igCIrmonhjb7yA/hVnApNgosFii1lC/wDyzX8qVbOEEEIvHtV3YKXYKVgsQpGEGAABUgFO20YFFgsJjioLhd0RFWu1RMtFgseS+Lf9GvTIRj5v616PoLb9ItW9UrgPiJFiPcP71dn4Tu0uNIto1OSiYNLqStzpO1JTj0ptUihc0ZpKKYxc0ZpKKAEo5pwANLgUXC4zmjmn4FGBRcLjOaDT8Ck20BcjxS0/bS7BTsIYBS4p22jFFgExQKdRgUWAKKKKLAFFFFFgCo8U+kpbANpaDRRcAzRmm0UXAqapdra2kjk9ia8Z1TUrjWtWESFtu4g4+td38Q9Rays41BxvGP1rlvAmnC5lM0g535/WoerJe52HhjQxaWqNJyw9a37iBTEw2jGPSriQqgwBUVziOF2P901Vh2PEfHYFjq0UqHG054r034eX7ah4bjmbrkV5R8QWkvtcghh53HBr1P4c2Mun+G44ZBgjHBqUtSUtTshS5pBS4FaWNAzRmjAowKLAGaQ0uBRRYBtFOwKMCiwDaKdgUYFFgGGinECm0tgCiiii4BRRRQIAMmq2pypBaOxOMLU5kCA9q43xXqchiaGI5LDGKQM4ab/iY+OoFByuCP1Fev2VikVsq4HHtXkvhywuodcju51xg9a9bW8DQBlNJEoo6/cQWOlzb9qkoccV5f4ZhXU9R84cgSEfrXRePJLu/ijitueMGs3wFp0+m5WZcMWzzS6h1PT7KHyIQg7VbWmRYZQak6U7FDuKQjikop2Cw3YKb901JTHHNKwWOQ8cyZ0mVc81h+CbBWmLN3FXvGtypMkBPPpVnwfFsjVh6UupPU6kgRwN6AV5L45k+2XbWynJYHivT9Tu1trSQk/w15IrnVvGkSqcqQR+ooYM0tD8MyRaWsgDA47VFp3hy6fVPMfzMI+ec16xYWCQWSREdKctnHGxKjk+1Ow7EdhEY7dVPYVoIKiRAOgqVeKY7EnamNThQQDQBAxPapoxxQUUCm+aqd6AFl4FZV/dLawF3OAKvSXSY5PArjPEt6bsvaQHJPShsTOb1G7e+u5QrEqPSuamt4jqUS5BJauoi0DVI4nkEYwVNca8dxFrkSzDB31BLPcPCsSw6SFHtW7WN4ayNMX8K2a0SLQUUUUgCiiigAooooAKM0UzvQA/tTWpw5oK8UAed+PbffZMcZwSap/DHU/tFzcwFv8AV8YP0Fdb4m04Xdi6+xry7wXNJpniO7R1KhpMD9Kl6Ml7nuxOVFNpsbbokb1AP6U6qKCiiigAooooActLSLS0AFFFFABR3oo700AUUUVQBRRRQAUUUUAFFFFABRRRQA00lK1JUsAooopAIaSnUYGKAPNvijzFZjrg/wBal8CKiW2QADmrvjfSm1G3RlyfLGa5XwzqU2nTeTLEVG48mpe5L3PXwwzWJ4iv0tbFzuGdpHWqd34hjgti8bB3/u5riL641DxHc+V5MkaA/ep3Hcy9CtTrviBbh1JSOUjke9e428EdvGEiAA9K5nwp4ej0i2OQGZuc4rp1600gSJRRQKKsYUUUUAFFFFABRRRQAUUUUAIabTjTalgFFFFIAooooAesasORmsq+0y1ldS8WSK142A7VFKQT0FAGbb6Xaqn+qq99mjWLAXipYmAXoKkLjbQBlzadbucmLNLFp1shBWLGKvlh6UZHoKAGIAowOlOoooAKKKKACo5Dg89BUlUtSfyrSRs4wKAPNfF9ysniFoRzmuy8OW4isI3xjiuAmtmv/E6zFiwr1XTIRFYRrjGBSQkcp4raeWBo4twyuK4rwvpF3Za3FdzhmCn0r2SS3if70an6impZ24bIhUfhRYLE9tL50CvzzTsU5UCrgDAo70xiDrTxTcU4UAKDSg02loAcwyKoyfxelXl5NNeIMG460AcD4p1o6epiUncw4Iqp4VhfUNlzLlmNdJqvhNNSk3tL+dXtG0NNKhVFbOKVtRW1ItSmWzsX4/hNeQCZLnxEjMhwJc5Ne3ajp4vYfLJxXLf8IFEJ/NEoBzQ0DR0uksn2NRHjHHStCs3TdONhEELlsVpVaGFFFFSAUUUUAFFFFABUf8VSVH/FQBKtOI4piVKQMUAZ97HvhYY6ivG/EIl0nXrdwu0STAn869slHBrzXx9p3mvbzKMhG3E0mJnoOn3SXFnCUOfkX+VXBXG+CL77ZYH5vuYX8uK7Ee1NDQtFFFABRRx60UAOWlpop3FABRRxRxQAUd6OKSmgFoooqgCiiigAooo4oAKKOKOKACijijigBrU3JpW9qbzUsBwopKWkAUuKQU7igCpNaiZWUjINYdx4Ut5pN2MHOeldRkCjcKAOTTwhADyWrYs9GgtV+UD8q1Nwo3CgBixADFO2AUu8elIWBpoBRRSA0vFUAUUcUcUAFFHFHFABRRxRxQAUUcUcUAIabSk0nFSwCiiikAUUUUAOBApGANNpQKAFUAd6UlcYzTSKaetADvl9ad2qMA07FABRRRQAUUUcUAFYPii/jtNLn3HDFeK3ScDJ6DmvKPibqjC9ht4m4cY/SkxMZ4PeS+u45SBgmvV4F2wha87+HtkYbSNmHPvXpCjihAiNhQBzT2pKYx46U3FO7UnFABRRxRxQAUUcUcUAG7HeneZnvTduaYR1oAl3imls9KiBp6qWoAMmlyO9PKHFRHg0AO60tNXpTuKsAoooqACiiigAooooAKAtFIGwaAHhcdKU0wNTs0ARSVz/AIks1n02Y45VDi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proof	Plane Geometry	Math	Chinese
48	"如图所示, 设 $A B C D$ 是矩形, 点 $E, F$ 分别是线段 $A D, B C$ 的中点, 点 $G$ 在线段 $E F$ 上, 点 $D, H$ 关于线段 $A G$ 的垂直平分线 $l$ 对称. 求证: $\angle H A B=3 \angle G A B$.

<image_1>"	['由 $E, F$ 分别是 $A D, B C$ 的中点, 得 $E F \\| A B \\perp A D$.\n\n设 $P$ 是 $E$ 关于 $l$ 的对称点, 则 $E P \\| A G \\perp l$, 故四边形 $A E P G$ 是等腰梯形.\n\n进而 $\\angle P A G=\\angle E G A=\\angle G A B, \\angle A P G=\\angle G E A$, 从而 $A P \\perp H G$.\n\n再由 $H P=D E=E A=P G$, 得 $\\angle H A P=\\angle P A G=\\angle G A B$.\n\n因此 $\\angle H A B=3 \\angle G A B$.']			['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proof	Plane Geometry	Math	Chinese
49	"如图, 圆 $C$ 与 $x$ 轴相切于点 $T(2,0)$, 与 $y$ 轴的正半轴相交于 $A 、 B$ 两点 $(A$ 在 $B$ 的上方)，且 $|A B|=3$.

<image_1>
设过点 $B$ 的直线 $l$ 与椭圆 $\frac{x^{2}}{8}+\frac{y^{2}}{4}=1$ 相交于 $P 、 Q$ 两点. 求证: 射线 $A B$ 平分 $\angle P A Q$."	['易知 $A C=\\frac{5}{2}$, 故圆方程为 $(x-2)^{2}+\\left(y-\\frac{5}{2}\\right)^{2}=\\frac{25}{4}$;\n\n可知 $A(0,4), B(0,1)$, 且直线 $P Q$ 的斜率必存在, 可设 $l_{P Q}: y=k x+1$.\n\n原命题等价于证明 $k_{A P}+k_{A Q}=0$.\n\n设 $P\\left(x_{1}, y_{1}\\right), Q\\left(x_{2}, y_{2}\\right)$, 将直线方程与椭圆方程联立得 $\\left(2 k^{2}+1\\right) x^{2}+4 k x-6=0$, 所以\n\n$$\nk_{A P}+k_{A Q}=\\frac{y_{1}-4}{x_{1}}+\\frac{y_{2}-4}{x_{2}}=\\frac{k x_{1}-3}{x_{1}}+\\frac{k x_{2}-3}{x_{2}}=2 k-3 \\cdot \\frac{x_{1}+x_{2}}{x_{1} x_{2}}=2 k-3 \\cdot \\frac{-4 k}{-6}=0\n$$']			['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proof	Plane Geometry	Math	Chinese
50	"如图, 在锐角 $\triangle A B C$ 中, $M$ 是 $B C$ 的中点, 圆 $O$ 过点 $A$ 且与直线 $B C$ 相切于点 $C$, 直线 $A M$ 与圆 $O$ 交于另一点 $D$, 直线 $B D$ 与圆 $O$ 交于另一点 $E$. 证明: $\angle E A C=\angle B A C$.

<image_1>"	['由圆幂定理知 $M C^{2}=M D \\cdot M A$, 所以 $B M^{2}=M D \\cdot M A$, 易知 $\\triangle A B M \\sim \\triangle B D M$, 所以 $\\angle A B C=\\angle B D M=\\angle A D M=\\angle A C E ;$ 又弦切角 $\\angle A C B=\\angle A E C$, 易知 $\\angle E A C=\\angle B A C$, 证毕!\n\n#']			['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proof	Plane Geometry	Math	Chinese
51	"已知函数 $f(x)=\frac{x}{\ln x}, g(x)=k(x-1), k \in \mathbf{R}$.

<image_1>
证明: 对任意 $k \in \mathbf{R}$, 直线 $y=g(x)$ 都不可能是曲线 $y=f(x)$ 的切线;"	['$f(x)$ 的定义域为 $(0,1) \\cup(1,+\\infty), f^{\\prime}(x)=\\frac{\\ln x-1}{(\\ln x)^{2}}$.\n\n假设存在 $x_{0} \\in(0,1) \\cup(1,+\\infty)$ 使 $f(x)$ 在点 $\\left(x_{0}, f\\left(x_{0}\\right)\\right)$ 处的切线方程为 $g(x)=k(x-1)$, 则\n\n$$\nk=f^{\\prime}\\left(x_{0}\\right)=\\frac{\\ln x_{0}-1}{\\left(\\ln x_{0}\\right)^{2}}\n$$\n\n切线方程为\n\n$$\ny=k\\left(x-x_{0}\\right)+f\\left(x_{0}\\right)=k x-\\frac{x_{0} \\ln x_{0}-x_{0}}{\\left(\\ln x_{0}\\right)^{2}}+\\frac{x_{0}}{\\ln x_{0}}=k x+\\frac{x_{0}}{\\left(\\ln x_{0}\\right)^{2}}\n$$\n\n对比系数得\n\n$$\n-k=\\frac{x_{0}}{\\left(\\ln x_{0}\\right)^{2}} \\Rightarrow \\frac{1-\\ln x_{0}}{\\left(\\ln x_{0}\\right)^{2}}=\\frac{x_{0}}{\\left(\\ln x_{0}\\right)^{2}} \\Rightarrow 1-\\ln x_{0}=x_{0}\n$$\n\n易知此方程有且只有一个实根 $x_{0}=1$, 此实根不在函数的定义域内，故假设不成立，证毕;']			['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proof	Elementary Functions	Math	Chinese
52	"如图, 在锐角 $\triangle A B C$ 中, $D, E$ 是边 $B C$ 上的点, $\triangle A B C, \triangle A B D, \triangle A D C$ 的外心分别为 $O, P, Q$, 

<image_1>
证明: $\triangle A P Q \sim \triangle A B C$;"	['连结 $P D, Q D$, 如下图.\n\n<img_4079>\n\n因为 $P, Q$ 分别为 $\\triangle A B D, \\triangle A D C$ 的外心, 所以 $P Q$ 为线段 $A D$ 的垂直平分线, 所以\n\n$$\n\\begin{aligned}\n& \\angle A P Q=\\frac{1}{2} \\angle A P D=\\angle A B D=\\angle A B C \\\\\n& \\angle A Q P=\\frac{1}{2} \\angle A Q D=\\angle A C D=\\angle A C B\n\\end{aligned}\n$$\n\n所以 $\\triangle A P Q \\sim \\triangle A B C$.']			['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proof	Plane Geometry	Math	Chinese
53	"如图, 在锐角 $\triangle A B C$ 中, $D, E$ 是边 $B C$ 上的点, $\triangle A B C, \triangle A B D, \triangle A D C$ 的外心分别为 $O, P, Q$, 

<image_1>
证明: 若 $E O \perp P Q$, 则 $Q O \perp P E$."	['连结 $O A, O B, O P, P B, Q C$, 延长 $O Q$ 与 $A C$ 相交于点 $F$, 如下图.\n\n<img_4079>\n\n由 $O, P, Q$ 分别为 $\\triangle A B C, \\triangle A B D, \\triangle A D C$的外心, 知 $O P, O Q, P Q$ 分别是线段 $A B, A C, A D$ 的垂直平分线, 所以\n\n$$\n\\angle A P B=\\angle A P D+\\angle B P D=2(\\angle A B D+\\angle B A D)=2 \\angle A D C=\\angle A Q C\n$$\n\n又\n\n$$\n\\angle O B P=\\angle O A P, \\angle A Q F=\\frac{1}{2} \\angle A Q C=\\frac{1}{2} \\angle A P B=\\angle A P O\n$$\n\n所以 $A, P, O, Q$ 四点共圆, $\\angle O A P=\\angle O Q P$.\n\n又 $E O \\perp P Q, D A \\perp P Q$, 所以 $E O \\| D A$\n\n$$\n\\angle O E C=\\angle A D C=\\frac{1}{2} \\angle A P B=\\angle B P O\n$$\n\n所以 $P, B, E, O$ 四点共圆, $\\angle O E P=\\angle O B P$.\n\n设 $E O, Q O$ 的延长线分别与 $P Q, P E$ 相交于 $M, N$, 则\n\n$$\n\\angle O E P=\\angle O B P=\\angle O A P=\\angle O Q P\n$$\n\n所以 $M, N, E, Q$ 四点共圆.\n\n又 $E O \\perp P Q$, 所以 $\\angle Q N E=\\angle Q M E=90^{\\circ}$, 所以 $Q O \\perp P E$.']			['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proof	Plane Geometry	Math	Chinese
54	"已知梯形 $A B C D$, 边 $C D 、 A B$ 分别为上、下底, 且 $\angle A D C=90^{\circ}$, 对角线 $A C \perp B D$, 过 $D$ 作 $D E \perp B C$于点 $E$.

<image_1>
证明: $A C^{2}=C D^{2}+A B \cdot C D$."	['由于 $\\angle A D C=90^{\\circ}$, 故 $A C^{2}=C D^{2}+A D^{2}$, 因为对角线 $A C \\perp B D$, 所以\n\n$$\n\\angle D C A=90^{\\circ}-\\angle B D C=\\angle A D B\n$$\n\n而 $\\angle A D C=90^{\\circ}=\\angle B A D$, 则 $\\triangle A C D \\sim \\triangle B D A$, 故\n\n$$\n\\frac{A D}{C D}=\\frac{A B}{A D} \\Rightarrow A D^{2}=A B \\cdot C D\n$$\n\n因此, 有\n\n$$\nA C^{2}=C D^{2}+A B \\cdot C D\n$$']			['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proof	Plane Geometry	Math	Chinese
55	"已知梯形 $A B C D$, 边 $C D 、 A B$ 分别为上、下底, 且 $\angle A D C=90^{\circ}$, 对角线 $A C \perp B D$, 过 $D$ 作 $D E \perp B C$于点 $E$.

<image_1>
证明:$\frac{A E}{B E}=\frac{A C \cdot C D}{A C^{2}-C D^{2}}$."	['由于 $\\angle A D C=90^{\\circ}$, 故 $A C^{2}-C D^{2}=A D^{2}$, 所以\n\n$$\n\\frac{A C \\cdot C D}{A C^{2}-C D^{2}}=\\frac{A C \\cdot C D}{A D^{2}}=\\frac{A C \\cdot C D}{A B \\cdot C D}=\\frac{A C}{A B}\n$$\n\n因为\n\n$$\n\\angle B A D+\\angle D E B=180^{\\circ}\n$$\n\n所以 $A 、 B 、 E 、 D$ 四点共圆, 故 $\\angle A E B=\\angle A D B$.\n\n由于 $\\angle B A C=90^{\\circ}-\\angle C A D=\\angle A D B$ 且 $\\angle A E B=\\angle B A C, \\angle E B A=\\angle A B C$, 则 $\\triangle A B E \\sim \\triangle C B A$, 故\n\n$$\n\\frac{A E}{B E}=\\frac{C A}{A B}\n$$']			['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proof	Plane Geometry	Math	Chinese
56	"设锐角 $\triangle A B C$ 边 $B C, C A, A B$ 上的垂足分别为 $D, E, F$, 直线 $E F$ 与 $\triangle A B C$ 的外接圆的一个交点为 $P$,直线 $B P$ 与 $D F$ 交于点 $Q$. 证明: $A P=A Q$.

<image_1>"	['<img_4223>\n\n如上图所示, 由于 $D, E, F$ 是垂足, 则 $\\angle B F C=\\angle B E C=90^{\\circ}$, 故 $B, F, E, C$ 四点共圆, 从而\n\n而\n\n$$\n\\left\\{\\begin{array}{l}\n\\angle B F D=\\angle F Q B+\\angle F B Q \\\\\n\\angle B C A=\\angle P C B+\\angle P C A \\quad \\Rightarrow \\angle F Q B=\\angle P C B=\\angle P A F \\\\\n\\angle F B Q=\\angle A B P=\\angle P C A\n\\end{array}\\right.\n$$\n\n故 $A, F, P, Q$ 四点共圆 $\\Rightarrow \\angle A Q P=\\angle A F E=\\angle A C B=\\angle A P Q \\Rightarrow A P=A Q$.']			['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proof	Plane Geometry	Math	Chinese
57	"如图, 设 $\triangle A B C$ 的外接圆为 $\odot O, \angle B A C$ 的角平分线与 $B C$ 交于点 $D, M$ 为 $B C$ 的中点, 若 $\triangle A D M$的外接圆 $\odot Z$ 分别与 $A B, A C$ 交于 $P, Q, N$ 为 $P Q$ 的中点. 证明: $M N \| A D$.

<image_1>"	['设 $A B=c, B C=a, A C=b$.\n\n在 $\\triangle A B C$ 中, $A D$ 为 $\\angle B A C$ 的平分线,\n\n$$\n\\therefore \\frac{B D}{D C}=\\frac{A B}{A C}, \\quad \\therefore \\quad \\frac{B D}{B D+D C}=\\frac{A B}{A B+A C}, \\quad \\therefore \\quad \\frac{B D}{a}=\\frac{c}{b+c}, \\quad \\therefore \\quad B D=\\frac{a c}{b+c}\n$$\n\n又 $B M=\\frac{a}{2}$, 由 $B P \\cdot B A=B D \\cdot B M$ 得 $B P=\\frac{B D \\cdot B M}{A B}=\\frac{a^{2}}{2(b+c)}$.\n\n由 $C Q \\cdot C A=C M \\cdot C D$ 得 $C Q=\\frac{a^{2}}{2(b+c)}$, 因此 $C Q=B P$.\n\n连接 $B Q, P C$, 并设 $X, Y$ 分别为 $B Q, P C$ 的中点, 如下图所示.\n\n<img_4233>\n\n易证 $X N$ 平行且等于 $M Y$, 所以四边形 $N X M Y$ 为平行四边形, 由 $C Q=B P$ 知 $N X=N Y$, 所以四边形 $N X M Y$ 为菱形, 从而 $M N$ 平分 $\\angle X N Y$.\n\n又 $A D$ 平分 $\\angle B A C, A B\\|N X, A C\\| N Y$, 所以 $M N \\| A D$.']			['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proof	Plane Geometry	Math	Chinese
58	"如图, 在圆内接四边形 $A B C D$ 中, 对角线 $A C$ 与 $B D$ 交于点 $P, \triangle A B D$ 与 $\triangle A B C$ 的内心分别为 $I_{1}$ 和 $I_{2}$, 直线 $I_{1} I_{2}$ 分别与 $A C, B D$ 交于点 $M, N$, 求证: $P M=P N$.

<image_1>"	['因为 $I_{1}, I_{2}$ 分别为 $\\triangle A B D$ 与 $\\triangle A B C$ 的内心, 所以\n\n$$\n\\begin{aligned}\n& \\angle I_{1} A B=\\frac{1}{2} \\angle D A B, \\angle I_{1} B A=\\frac{1}{2} \\angle D B A, \\\\\n& \\angle I_{2} A B=\\frac{1}{2} \\angle C A B, \\angle I_{2} B A=\\frac{1}{2} \\angle C B A\n\\end{aligned}\n$$\n\n故\n\n$$\n\\angle A I_{1} B=180^{\\circ}-\\frac{1}{2}(\\angle D A B+\\angle D B A), \\quad \\angle A I_{2} B=180^{\\circ}-\\frac{1}{2}(\\angle C A B+\\angle C B A)\n$$\n\n在 $\\triangle A B D$ 与 $\\triangle A B C$ 中, $\\angle A D B=\\angle A C B$, 所以\n\n$$\n\\angle D A B+\\angle D B A=\\angle C A B+\\angle C B A\n$$\n\n因此 $\\angle I_{1} I_{2} A=\\angle I_{1} B A=\\frac{1}{2} \\angle D B A$, 则\n\n$$\n\\angle P M N=\\angle M A I_{2}+\\angle M I_{2} A=\\frac{1}{2}(\\angle C A B+\\angle D B A)\n$$\n\n同理 $\\angle P N M=\\frac{1}{2}(\\angle C A B+\\angle D B A)$, 所以 $\\angle P M N=\\angle P N M$, 即 $P M=P N$.']			['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68htm3G6v1soD1G1SIw304z+JoA98+C/huLRfh3YXTW6Le34NzI5X5irH5Bn02bT+Jr0jIrMur7SfDWjxtd3VtY2FsixI00gRVAHyqM9TgdKpeGPF+j+L4byfRZ3uILWbyWlaMqGOM/LnnHPegDoKKKKACiiigAooooAKKKKAPA/2kL20SDRbARR/a3eSd3AG5UA2gZ9yT1/u10HwD8NQaf4IGsTW6fa9Qmd0lK/MIl+QAHqBkMfxryT4x6q/iD4qX1vbkyi1KWEKr1yv3h/32zivpA3ukfDzwTZDU7pLezsYIrfeFJ3sFA4UckkgnH1oA6fNFcT4U+KXhjxlqLadptzMl3sLLDcR7C4A528kH6deK7agArjPHXgn/AISSCHUtNl+w+IrA+ZY3qcHIOfLb1Q+/T6Eg9nRQByPgvxiviaCe0voDYa9YMI7+xfqjdmX1Ru3pn6E9dkVwvjnwZd6nJBr/AIamWx8TWI/czDhbhP8Ank/qD2z0+hNaXgvxfb+K9OkLRm01S0byr+xcYeCUcEYPODg4Pt60AdRRRkUUAFFFFABRRRQAUUUUAFcF8WtLurzwf/amnErqeizrqFuyjJ+T7w+m0k474Fd7TGUOpVlBUjBBHBH9aAM/w/rMHiHQLHV7bHlXcCygZzjI5X6g5H4Vp15v8NSdC1nxF4KlbC6ddm5sVP8Az7S/MAP90nB92r0igAooooAKKKKACiiigAooooAKKKKACiiigDz34dHyfFXj207LrHn4/wCui5/pXoVeeeDf3fxV+IcPpJYyAf70LGvQ6ACiiigBaSlpKAOU+JjbPhn4iP8A05SD8xitPwouzwfoif3bCAf+Q1rK+J4z8MvEQ/6c3Na/hc58JaMfWxgP/kNaANaiiigAooooAKKKKACiiigArzOxjbXPj7qd6Rut9B02O1VvSWTLcf8AAWcH6V6ZmvO/hN/p2ma74jbltZ1aeaNvWJTtQfhhqAPRKKKKACiiigAooooAKM8ZorjPiH4oudB0iCz0lRNr2qSfZrCEdQx6yH2UHP1x2oAxPFc7+PPGEPgmxlY6TaET67NE3UAkpBn1JGSP6qRXpMMEdvCkEMaxxRqFRFGAoHAAFc/4K8I2vg3QFsIWae4kYzXdy/LTynG5jn9B2Hqck9LQAUUUUAFFFFAHP+NNej8N+DdV1VpUSSC3fyc95SMIPxJFfN/wK0lNV+JsE82GFjbyXWG5y3CKfrl8/hX01rHhvR/EKQpq+nW96kJJjWZM7ScZx+VZ9t8P/CVnOlxbeHrCGZDuV44grA/h70AXPE8SN4W1YtGuRZXABIHGUOcH3rzD9m7/AJFPVx/0/D/0WteoeJzN/wAIzqUMFtPczzWskUcUK5ZmZSAOwxz3rzz4H6DrfhjStS0zWtJubN5pxPHI4DI/ygEcHgjA+tAHrlFFFABRRRQAUUUUAFZ+s6rb6Hot7ql26pBawtK27vgE4HuelaFZuraFpevWq2+rWEN5ErbhHKu5Q3rQB8qfCzTn8T/FjT5rxg5Wd7+YtzuZcuPrlsV9XXOl2d9c2tzdW6zSWjM8G7kIx4LY6Zxxk9Mnpk1jR/DrwfDIskPhzT45FOVdIQCD7Yqr8Qr/AMS2Xhtrfwppc13qNyTGJUIAtl7vyeW5wPz6DBAOE07QI9Y/aMvtV06FY7DSAv2mWHhXuDHtKcd8k5/3TnrXtteJeCT8SbO60rRJvDVtpGjrcrJdXUK5lkAO5i7NI2S5GCcZ54x29toAKKKKACuA8b+Fb+K/j8YeFF8vxBaKBNCPu38IxmNx3OBwevAHXaR39FAHN+D/ABbYeMNFW+tAYbiNvKurV+JLeTurf0P9a6SvOvGPhzUNH1VvG3hSE/2pEoF/YqPl1CH0wP4x2xycfn1XhjxLp3ivQ4NU06UPHIPnTI3RP3Rh2I/Uc0AbdFFFABkZxmjIxnNcH8SPFWveC9FbWrGLTbizR1jeGcOsiluAQQ2G5xxgcZrE+Fnxan8d6veaZqNpbWlwkPnQeSxxIAcMDuJ55B496APV6MisHxdqOp6N4ZvtU0xbRpLKF7mSO5ViHRFLMAVIweOKzvA2reJdf0e01jV4NPtbe7i82KCBHMm0/dZmJwARk4APBHPWgDr6KKKAPNvHZHh3xv4Y8YJhYGl/svUGzx5MmShPsGyc/SvSMj1rE8X6BH4n8J6lo0gH+kwsIyeiyDlD+DAH8Kxvhbr9xr3ge0N9v/tGxZrK7WT7wkjOPm99pUn3oA7WiiigAooooAKKKKACiiigAooooAKKKKAPPfC2F+Mvj4f3otPbH/bEivQq888Nf8lr8ce1vYg/9+69DoAKKKKAFpKWkoA5n4iR+b8OPEi+mnTt+SE/0qz4Mk83wL4fk/vaZbN+ca1J4uh+0+DNdgAz5mnzrj6xtWd8Np/P+Gvh1wc4sYk/75G3+lAHVUUUUAFFFFABRRRQAUUUUAc5491b+xPAet6iH2vFaOI2Bxh2G1f/AB5hR4D0kaJ4E0TT9u14rRDIP+mjDc//AI8TXPfFrN/pei+HFyTrWqwwSKP+eStuc/otehAAAAcAY4AoAdRUUrOkbvHH5jhSQucbj2Ga8u1/44WPhnV5tK1Xw/qMV3EFLqssTcMu4chvSgD1ais7RNVh1zQ7DVbcFYryBJ1VjkqGAOCfbOKyvGvjPTfA2iDUtRDybnEcVvHjfKSedoJxwMkn0HuKAOmorzjwl8V18bXk0GjeHNQdYADLPLJGqR5PGTnvg8Dng16PmgChrGrWWhaTc6pqEwitLZC8jkZwB2A7knj61w/gPTrrxFq83xA1qIpNdr5ek2r8/ZrXPDf7zdc+hPY4qrq3/FzfGbaFGxPhfRZA2oyKcLeXH8MIP91ep/8A2TXp0caRRpHGiqiqFVVGAAOgA7UAPooooAKKKKACiiigAoprOiKzM6qqgkknAAHWqWma1pesxSSaXqVpfJG212tplkCn0O0nFAF+ijNU59X021z9o1G0hx/z0mVf5mgC5RWBceN/CltnzvEujoR2N7Hn8t1Zs/xV8DW+d/iWyP8A1zLP/wCgg0AdjRXnz/GrwGG2w6xJcP2WKzmJP5rTT8YNFk/49NF8R3np9n01jn8yKAPQ6K88/wCFnX83/Hp8PvFj+89mIh+pNL/wnHjSf/j0+Gl6fT7RqMUP8waAPQqK89/t/wCJ0/8AqvBemWv/AF8amsn/AKBSGX4tzDIh8H2i9zI9wx/SgD0Ok3AkjIyOozXgmp+OPiC+pHS9E1rTNY1IEBodIsDLHFnu8r/KP198Vza+BPiro2v3XiPyrs3ZzJPcWtxHJJKG+8FTPzEZPy4xxx2oA+oKK8d8MaSfGlo81v8AEvxI00ZxPaIy200DdMMg6fhxW/8A8KksZv8Aj88VeLLz1E2qEj9FoA9DJwMngVm3XiLRLHP2vWNPt8dfNukT+Zrjh8FfBb/8fVpe3fr599Kf5MKuQfCLwFb42eHLc/8AXSSR/wD0JjQBdn+JPgu24k8T6Wf+udwr/wDoOazZ/jJ4At/v+IY2PpHbyuP0SteD4feDrcYTwtpH1ezjb+YNaUHh3RLXH2fR9Phx/wA87ZF/kKAOIb46eCi/l2099dt2WCzck/mBXm3iXxunhzXW8T+DdK1jS2viY76PULHZaXB5KuPm4fIPTryf72fpBIkjUIiKqjsBiq2o6XY6xZtaalZwXduxyYp4w657HBoA5v4b+LLnxp4Ng1a7t0huC7xSCMfIxXuM5wOeldhXleraHqHw21SbxH4XgM3h+Qh9T0WIfc7GWEdBjqRxwPT7vomj6vYa7pVvqem3CT2k67kdf1BHYg8EUAcf4yjTXvHnhXw0ypJbwvJq14jcjbGNsYI6EF2P5V4H4gs7n4T/ABaEtlnybeYXNsM/fgfOUP4bkP0Jr3/wpjWviL4r8QNl4rV49HtW9BGN0o/77YflXO/HzwgdY8Lpr9rFm70vPm7RkvAfvf8AfJwfYbqAN74haxFqvgSytNMnDf8ACSz29nbyL12SnLN7fID+dd5b28drbQ28CBIokVEUdAAAAPyFfOfwQub/AMRa1pun3RZtO8OpcXURPTfLtVVP0zIw/GvpGgAooooAK84sR/winxlu7L7mn+J7f7TCOgW6iHzgfVfmPuRXo9cF8VtPuG8Nwa/YLu1HQLhNQi/2kU/vFPsV5P8Au0Ad7ketFU9M1C31fSrTUrVt1vdQpNGc/wALDP8AWrlABRRRQAUUUUAFFFFABRRRQAUUUUAeeeE/3nxf+IMvXaNPQf8Afk/4V6HXnvgH9945+IF11DajFDn/AHEI/rXoVABRRRQAtJS0lAEF3ALqynt26Sxsh+hBFcT8GZzP8K9GDffi82Jh6FZXGPyxXe1558Ix9m0jX9LbhtP1y6gA/wBnIIP45NAHodFFFABRRRQAUUUUAFFFFAHnOpuNa+OOiWS4eLQ9PmvZPQSS4QA++MEV6NXmHwzzq/jLxx4lPzJNqAsYG9UiGMj6gpXp9ABXx78SFk1O9Xxa7M0etXl19mbsYISkaceuB/KvpT4k68fD/gPUrqIFrqaP7NbKOCZJPlGPcZLf8BryP42eH10T4f8Agy0CjNgrWrNn+Iopb65KZoA9N+GOq26fB7R9Qu51itra1fzJX4VFjdlP5ba8M+Les6j4gn07V7/MFtdrI+mWWMGO1BGJH/2pDz9FHXPHT/DIXfjbwrpvhAJImi6fO1zqsx6TL5heOBfYnLH2HrXF/Eu/l8XfFi6s7M71W4TTbSMcgbTswPYuWP40Ae5/A7Qho/w2tJ2j2z6g7XUhI5wThPw2gH/gVY/j34zWWmXOoeGtFEi6qr/ZjfS4WC3cnDNnqSvPbGR37+i3t9pvgnwj9ouX8uw021WMYAyQoCqoHqeABXmnhj4UW/iS/HjXxNFJFfX1y16umR7ViRScqsmRlj3IyOuD3oA0vDPjr4b+CfDltpMHiOKTyxummWKVzLIeXc7VJ5P9K1P+F0eD5f8Aj0l1C89Ps9hKf5gV29vplha/8e9jbw/9c4lX+Qq1j17UAeej4s20vFp4Q8XXR7GPSyR+rUp+Imvzf8enw519vTz9kP8APNeg9etLQB57/wAJb8Qrgf6P8NxGv96fWIR+mM0f2n8VZ/8AVeHvD1r/ANfF48n/AKBXoVGaAPPfs/xbuP8AWXnhK0B/54xTuR/31R/wj3xNuP8AW+OtPtc9fI0pHx/30a76eeG2geeeWOKKNdzySMFVR6knoK8yvvEer/Ea5n0bwbI9noy/u73XmXG4HqkA6k/7XGOenBIBx/iaPxRqd+/hLSvHF74h1KYGO7ht7SOC3t4+jea6k49No56j0BztO+Cvj/RLaa5sdVjhkZgJLSxvnge4Qdt+0KD6Z4/GvdfDHhPSfCOkLp2k2/lIOZJW5kmbuzt1J/l0GBxW7QB872ei+AZbxbHxoniPSdT6bNYvC0Tn/ZlAAI9+K9Csfg18O2gSeHR1uUYZWQ3srq35Pg/lXdX+mWOq2jWuoWcF1bt1injDr+RyK4Bvhjf+G7l73wHrsumluW068JmtZPb+8v15PuKANyD4XeB7f7nhmwP/AF0Qv/6ETWpbeD/DNkwNr4e0mE/3o7KNT+YFcmnxF1Tw6yw+OfDs+nx9P7TsQZ7U+5xlk+hya7fSdb0vXbQXOl39tdwnq0EgbHscHg0AXI4khQJFGkaeiAAflT6XIooAB07YorkfEnxD0Lw3crYu8t9qz8RadYp5s7H0wOn4mufk0bxt4/jK61MfDOhvybG0fddTL6O/RR7AfUUAbOvfEvR9JvP7M05Zdb1lshLDTh5jA/7bDIUD8T7Vjt4U8ZeN1z4v1RdJ0mQfNpGmN87j0kl5z7gZHsK7Hw74S0XwnYfY9FsUtkPLuOZJD6sx5P8AnFblAGVofh7SvDWnLYaPYQ2luOSEHLH1Ynkn3Oa1aKKAOI8TfD6LU9TGv6Fdvo3iOMfLeQjKTf7MqdGHGM/TrjFUtJ8f3elahFonjuzj0rUH+WC+Q/6HdY/uufuH2P6dK9ErN1rRNP8AEGlz6dqdrHc2sw+ZHHfswPUH0I6UAaIIIyCMetLXlcVh4q+GABsTceJPC6Hm1b/j7s0/6Z/31Hpx9B1rvfD3ibSPFOmrf6RepcQg4YA4aM+jL1B9jQBr0UhIHUiloAKKKKAGlQwIYZB7Y/CvItftNb+E8uoax4XskvfDl1uluLBiQLKXH+sTH8HAyPbHAxj1+mPGsiMjqrKwwQwyCO4oA8++Elpq1h4VSK+WyuIrmSS8S/tZw4nMjAkkbRg8n8sV3l7aQahZT2dzGJIJ42ilQ/xKQQR+teXss3wg1p5Qkk3gm/n3MFBZ9Mmb27xn/wCt1HzepW11Be2sVzbTJNBKgeORGBDA9CCOooA4P4W/DyXwFBrKXE0c0l3d/unQ5zAg+TdwMN8zZA46Yr0KiigAooooAKint4rq3lt50DxSqUdT0IIII/I1LRQB5p8KryfTH1rwPfMWuNBuMWzN1e2c7kP4Z/JlFel15v40A8M/ELw54tHy2tyTpOoMP7jnMbH2DdT6ACvSKACiiigAooooAKKKKACiiigAooqOaZLeF5XOERSzH0AoA4H4VHz08W3/AFF14iu2Q+qAqB/WvQq8/wDgxG//AArSxuZBiS8mnuWHuZWH8gK9AoAKKKKAFpKWkoAK888FH7D8T/HmlnhXmtr2Meu+P5j+eK9Drze+mGj/AB905yMRa1pD25PYyRMXB/75AH40AekUUUUAFFFFABRRRQAVm+INUXRfDupamxAFpayT89yFJA/MVpVwHxinc+Bf7KgYi41e9t7CLHUlnBI/JSPxoAt/CnSv7J+GuixsuJbiH7VKT1ZpDvyffBA/Cu0qG3gjtbaK3hXbFEgRFHZRgAfkKmoA8o+Ilzq2p+LvD1jD4a1W70fTL9Ly7mjttyysPu7OfmAy2fXPtVr4weHdQ8Z+DNMt9Hs3kuW1CKTbJGUZEKOp3ZGVALDOemK9NooA4GxsI/hn4JTStI0u/wBTvxC0h+y2rETTkY3O3QDOOCSQo78Z8U+H/g7xJpHxC0zV9c8Oar9khleWVxbMxDbG2nA5+/tNfVNeffEXWb2d7PwXoEm3WNYOJZFH/Hraj78h9Mjgfj3xQBkrLJ8U/G8bRqT4P0KbdvZTi+ugOBg9VXP+d3HqwGKzdB0Sz8O6La6Vp8Xl21sgRB3PqSe5JyT7k1p0AFFFFABRRRketABWVr3iHS/DOlvqOr3aW1snGWPLN/dUDkn2FZHi/wAdab4SiSB0e71W44tNOtxulmY9OB0Hv+WaxfDvgi/1jUY/E3jspdaoMm104YNvYqewXJDN7n9cAgAzotN1j4q3kV5q8VxpXg+N/Mg09jtmv8HIaXHRT6fl/er06zs7fT7SO1s7eO3t4l2xxRIFVR7AVOBTqACiiigAoooyKAGOquhVlBUjkEZBrhNb+FOjXlz/AGhoM0/h3VVJIutOOxW9mjGAR9MZ75rofEXi/Q/ClqJ9Y1COAtzHF1lk/wB1Byfr0rilv/Hvj2YpZW8nhTQWJ/0udQbyVP8AZU/cz68Y9TQBRuviH4j+H19b6X4tSx1tJCFS402ZVuiD0LwHGSfbA9zVSx8T678WbyWz03W7bwzpiOyPbpMGv5h3OONo+hH/AAKvQvDfgDw/4Xf7RZWZlv2yZL+6PmzyE9SWPTPfGKXxD4A8N+KCZdS0yM3XBW7hJjmUjod64Jx2zmgCbwx4N0PwjaeVpVmElYfvrqT5ppj6s/U+uOnsK6LNeYzaT8QvBhVtEvx4p0pTk2eosFuox/sy/wAX4/gK1dD+KOh6jdLp2qibQtXAAey1JfKOf9lzhWHp0J9KAO5opMgjII9aXIoAKKKKACiiigBCM/1964DxR8N/tmqHxD4WvDoniFessX+pufaVcc/XBz3B7egUUAcFoPxDT7euheLrT+w9c/hWRsW9z/tRSdPfBOewJrvc1k674c0nxLpzWOsWMV1AeQHGCp9VYcqfcYrgDN4m+Fs6ib7Tr/g0f8tcb7uwHv03oPX+XQgHqtFZmia9pXiPTU1DSL2K7tm/jQnKn0IPKn2NaeR60AFFFFAFe6tYby1ktrmFJoJVKSRSLuV1PUEdDkV5haST/CTWzYXbvL4Lv5s2twxLf2bKesb/AOwex/HrnPq9VNR0601bT57C/t0ntZ0KSRP0YGgCyjrIoZGDKRkEHIIp1eU6NeXnwv16Lw/rN1JceFr1sabqE5ybWTtDIew9CfbpyB6tmgAooooAKKKKAOe8beHx4o8G6ppAUGWeE+TngCRfmT9QKr/D3xAfEngfTb6Un7WkfkXSnqJk+VsjsSRn8a6mvMtBm/4RT4u6x4cddlhrq/2pYjsJsESqPrgn2Cj1oA9NooooAKKKKACiiigAooooAK57x3fjTfAWv3WdpSwlCH/aKkD9SK6GvPvjJI8ngMaXGxEuq31vZJjrkvuP6IaAN3wFZHTvh/4ftWXay2MRcf7TLk/qTXSVHHEsUSRRjaiKFAHYAYFSUAFFFFAC0lLSUAFeY/FtTpt34S8SqDjTNWRZWH8MUmA2fb5QPxr06uX+ImkDXPh9rlhjc7WryRjuXT51A/4EoH40AdRRWD4K1f8At3wXo2ps2ZLi0jaQ/wC2AA//AI8DW9QAUUUUAFFFFABXnfiwjVviv4N0c/NFZifU51+g2xn/AL7z+deiV514a/4m/wAYvFurH5o9Pt4NMhb/AMfkH4MP1oA9FooooAKKKa7pGhd2VVUEkscACgDD8XeJbXwl4autYufn8pdsMQ6yyH7qD6nH4ZrD+HnhO50qC48Q62TL4i1gia7Zlx5KnlYgOwAwD7gDoKx9FP8Awszxq2vyZbw1ocpj02JlwLm4H3piD1C/w/h0O4V6lQAUUUUAFFFRzTw28LzTypFFGpZ3dgqqB1JJ6AUASZrzzxP4+uW1c+F/BkCajrznE0x5gsVzgtIR1I9PX1PynOk8Tav8TbybS/CUsunaDEdl7rTLiSXPBSAHp0+91Ge3Ge68OeGNJ8KaYlhpFoIIxje55eVv7zt1Y/Xp0GBxQBl+EvAtp4bkl1G6mfU9duebnUrgZds/woP4V9v8AB11FFABRRRmgAorP1fWNP0LT5L/AFS8itbWMfNJK2Bn2Hcn0FcBH408VeON8XgnSlsdNJwda1NcA+8cXO72JyPUCgDudc8R6N4bsjd6zqMFnD/CZG+ZyOygcsfoK4SLxF4x+IDGPw1aHw/obddWvo900w/6ZR9B9f1BrZ0T4Y6RYXn9qaxLNr+snBa91H58EdNiHIUDt1I7Gu2xwOPyoA5Hw78ONE8P3R1CRZtU1hjufUdQbzZc/wCznhfqOfUmuvAx7YpaKACiiigArL1nQNK8QWZtNX06C8hwcLKmdue6nqD7jBrUooA83Pw61nw0fN8D+I57WJTkaXqJM9qfZc/Mn1GT71HafFKfSLtLHx1oF1oUpO0Xqgy2jt7Mo4z+OO5FemVDcW8V1A9vcQpLC4wySKGVh6EGgBtjf2epWiXVhdQ3Nu/3ZYZA6n6EcVYzXn958KdOgu3v/C2pXvhu9fk/YmzA5/2oTwR7DA9qozeLvGPgpR/wlmhrq2nrwdU0gZI95IjjH6D0zQB6dRWD4c8YaF4ss/tOjahFcAcvHnEkf+8p5H16VvUAFFFFABTSMg/T+dOooA8+1v4eS2uovr3gq6GjaweZYVX/AEW79pEHQn+8Oep681L4V+I0eqaidC8QWL6J4gTANrOcLN/tRMfvA+n5E13lYfiTwno/izTvsWrWgmVcmKUHEkR9VbqD/PvmgDczRketeVvrHin4ZEJriy+IPDCttXUo+bq1Xt5o/jHv9eegr0bStUsdb02HUNNuY7m0mG9JUOQf8CD27UAXqKM0UAZ2saNY6/pU+malbrPaTqVdGGMe4PYjsfWvP/D2t6l4E8QweDvEsrTaXN8mkavLxvA6QyHpuA4H0A6EY9RrI8QeHtP8UaPPpeqQCW3l79GjYdGU9mHr/iQQDXyKK828MeJL/wAM6/H4I8WT+ZOw/wCJVqbcLeRdlY9pB09/xBb0mgAooooAK89+LGnSR6HaeKbGMnUvD9wt4mOrxZHmIfbHJ/3T616FUF3bQ3tpNa3EYkhmQxyKehUjBH60ANsbyHUtPtr62ffBcRLLE47qwyDVmvPfhVczWemap4Tu5C9z4eu2t1ZurwOS0TfiN34AV6FQAUUUUAFFFFABRRRQAV5x4zkXVfih4J0HOVgll1OYf3fLU+Wf++gwr0evMfCoOufGfxdrZy0OmwxaXAx6Z6yAfRlP/fVAHp1FFFABRRRQAtJS0lABTSoZSGAIIwcjr606igDzr4Tn+zLbX/CrkhtF1ORIgf8An3kO+M/j81ei15zcf8U58b7a4Py2viSwMLHoPtEPIJ/4Bx+NejUAFFFFABRRRQBWvbuKwsLm8mP7q3iaRz/sqMmuE+DMU0vgiXWrvm61m/nv5T7ltv5fLkfWrvxZ1J9P+G2rLBzPeKtnEo6sZWCkf98lq6Lw5o6aB4b03SUORZ2yQlh/EQOT+JyaANWiiigArzfx3qF34k1y3+H+jTtDJcp52r3KHm3tePlz/eccY9CM8Gug8beLrfwdoLXkiG4vJ38mztE5aeVui4649f8AEiqvgHwrc6Bps9/qrifX9Uk+06hMecMeRGP9lc/nnHHFAHSaTpVnoemW2m6fAIbW3QJGg7f4n1NXqKKACijI9a5Hxb470/wq8discuo63c8Wum23MkpPQnH3R7n0OAcUAa3iLxHpXhXSX1LWLlYLdSAMjLO3oo7muEg0rWvijNHfa8k2l+FNwe30pGKzXg6h5iOinsP5YDHQ0PwNfatqsfiTxzJHeakvNrpyDNtYj0A/if3/AJ4BHoAGBgDHtigCK0tLewtYrWzt44IIl2pHGoVVHsKsUUZHrQAUVDc3VvZW0lxdTxQQRjc8srhVUepJ4FefXHxGvdfunsfAWjtq0iHa+pXGY7OE/wC8eXPqBj2zQB3epanY6PYSX2oXUNtaxDLSysFUZ6de9eft8QNY8XyyWvgDShNCDsfWb9THbof9lT8zn27HqCKsWfw1k1W8TUvHOqSa7dqdyWfMdnAfRYx976nqOorvre2htbdIIIY4YY12pHGoVVHoAOlAHC6Z8LrR75dU8VahceJNTHzA3nFvE3fZD90D26ewrvlVUUKihVAwAowBTqKACiiigAooooAKKKKACiiigAooooAKbzjpTqKAOP1/4a+HdeuftwtpNP1QHcl/p7mGZW/vEjhj7kE+9Y/kfEbwhzDNB4u01f8AlnLiC8QezfdbHvkntivSKKAOI0D4o+G9cvf7Nlmm0zVVbY1lqKeTJu/ug9CfbOfau3rE8QeEtC8UweTrOl291xgOy4dR7OMMPwIrkf8AhDvGXhf5vCXiU31mnK6ZrQ8wY9FlHzD2HAoA9Jorzu2+KdvYXC2XjLSbvw5dnhXmUy28h/2ZV4P8veu9tLy11C2S5s7mG4t5BlJYXDqw9QRwaAJ6KKKAGMiupVlBUjBBHBFed6j8PtQ0PUJta8AXq6bdSHfcaZMM2lyfTH8B+n0G0V6PRQBxXhbx/a63fvompWr6R4jhOJtPuD944yWjb+MY59ce3Ndrmub8U+CdH8WwL9vheO7i5t723bZPCevyt9ex4z2rlIvEPiP4dyi28WiTVtBzti1yCMmSEek6Dn/gQz+JJoA9Poqrp+o2Wq2Ud5YXUNzbSDKywuGUj6irVAGF4o8K6d4t0l7DUYiCPmhuEOJIH7Mh7EHH1rm/CHifUbHWD4K8WSD+2ok3Wd3/AA6hDzhh/tjByO+D6GvQa5jxl4NtPGGmxxSSyWl/bP5tlfxcSW8gxyO+DgZGR0HQgEAHT5HrRXB+DPFt7JqEnhTxQq2/iO0XKyDhL6LtKnqcDkD0+oXvKACiiigDzTxJIPCHxW0fxCfl0/W0/su8I6LNnMTn6/d9gpr0uuW+IPhceLfBl9pi8XQHnWrjqsy5K89s9M9gTUngPxF/wlHgzTdTc/6S0fl3IPBWVeHyO3Iz9DQB0tFFFABRRRQAUUUUAVNTv4dK0m71Cc4htYXnk/3VBY/oK474R2Ett4Ct7+6X/TdWmk1Gdv7zSMcH8V2n8ab8W7yY+EYtDs2xe65dxafFjsHbLE+20EH6121lZw6fY29nbrtht41ijX0UAAfyoAsUUUUAFFFFAC0lLSUAFFFFAHn/AMXrC5fwlHrtgP8AT9Buo9Rh4zkKfmB9sHcf92ux0fU4Na0Wy1S2P7i7gWZfYMAcH3GcVanhjuYJIJo1khkUo6MMhgeoP4GvP/hVK+lw614NuXJn0K8ZYd3VreQl42/U/TigD0WiiigAooooA87+IX/E18W+CfDo5WbUWv5h/sQLnB9iSfyr0SvO7L/ib/HbU7gfNDomlR2o9Fllbfn67eK9EoAKinnitreS4nkWKGJS7uxwFUDJJPYAVLXl/jS+uvG3iVfAOks62aFZtbvEPEUYORCD/eOBn8B2agBfCsD/ABA8Wt431CJhpFkWg0K3kHBGfnuCPUkcfT/ZBr0+q9paQ2NrDa20SxW8KLHHGowEUDAA9gKsUAFFQ3V1b2dtLc3U0cMESl3kkYKqqOpJPQV5hea7rXxOZ9M8KmfTvDocrd626lXnA6pCp7HuePfHQgGlr3j2e61aTw14Jt01LXOk9yf+PayHQs7dyP7o7+p+U6nhDwJa+GZJ9Su531LXrv5rzUZvvOf7qj+FRgcD0HoANjQPDmm+GNIh0zSbVbe3jwSR952xyzHuff8ApWvQAUUmQASSMCuE1j4m2EV82k+HLSbxDrPTyLP/AFUZ/wBuT7qj86AO5lljgieWWRI40BZndsBQO5PavPr74oRX13JpngvTZvEOoKdrSxfJaxH1aU8Y+nB9aqp4B8QeLrtL7x9qoe0Vg0eh2DssC+m9urH+R6NjivRLKwtNMs47OwtYra2jGEjhQKq/QCgDz5PhtqHie4jvviBq76iyNvj0uzJitIj6f3nPucHtkivQrKyttOs4rSzt44LeJdscUahVUewqzRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAVr2wtdSs3tL+2hubaQYeKZA6t9Qa4GX4WtotzJfeB9audBuHO57Rv31rKfdG6H35x2xXo9FAHm3/Cc+JfC2Y/G3htjarwdV0jM0OPV0PzKPc/lXY6H4m0TxJb+do2qW16oGSI3y6/7yn5l/EVrY46dq4nxD8L9A1qb7bZRPo2rKdyX+mnyXDepxgH3PX3oA7iivM47z4jeDlMd/Yx+LNNTpc2h8u7VfdD98+w5966Dw98Q/DXiOT7Pa6gLa+3bXsbweTOrf3dp6n/dJoA6yoZYI7mF4poleN12ujqCGHoQeo9qmyPWigDzS68Cat4OuZdU+HsypG7b7nQ7lyYJ/eMn7jfj6DIAxW54W8f6V4mnbT3Emna3FkT6ZdjZMhHXbnhh7j8QK6+uX8WeB9K8X2gF3Ebe/iwbbUIPlngYcghupAPY8fQ8gA6jNFeYWvjHWvAskWl+O4nuLLO231+2QtG49JlHKsPXnPvgk+lWt3b3trHc2k0c8Ei7kkjYMrD1BHBFAHMeNfBqeKLW3uLWc2etWD+dYXq9Y34+U+qnAzVbwV4yuNWnuNA8QW62PiaxH7+36LOvaWP1U9/Su1rjfHfguTxHb2+oaVcfYvEOnN5ljdrxz3Rv9k/ofUEggHZUVx/grxtF4oS4sryA2Gv2B2X1jJwyHpuT1Q/pkZ7E9hketABXm/hT/AIpf4m+IfDDfLZ6jjV7AdstxKo/EZA9Fr0ivOvilHJpY0PxlboTJoV6DcBerW0mEkH8vpkmgD0Wiobe4iureK5gkWSGVA8cinIZSMgj1FTUAFFFFABRkUVR1fU7fRtHvNTum2wWkLzP9FGcD34xQBw1x/wAVN8bbaAHdZeGLMzSHt9pn4A/74APsQa9Hrz34Q6Td23hWfW9SX/iY67dPfyk9QjfcH0xyP96vQqACiiigAooooAWkpaSgAooooAK828Wf8Ut8TfD3igfJZaip0nUHHAG75omP/Aup7Ba9JrA8ZeHk8U+EdR0dhh54iYXPGyUfMh/76AoA380Vxvw18USeKfCEMt4Nup2jm0vkPBEqcZI9xg/XI7V2VABRketFc9451oeH/A+s6nv2NDav5R/6aMNqf+PFaAOc+FP/ABMIfEniM8/2rrEzQt6wJ8qfl81eiZ5xXL/DzS/7E+HuhWO3a62aSSL6O43t/wCPMa3r+/ttMsLi9vZVitrdDJK7HhQOSaAOc8e+KW8M6CPsiefrN8/2bTbdfmMkzcA4PZSc8+w71L4F8KDwp4eS1lYTajOxuL+6PLTTtyxJ7gZwD7Z6k1zHgOwvvFuvy+P9bieONwYtDtXA/cW54L4/vMOM+mccEY9PyD3oAMj1rH8R+JNK8LaU+o6tcpBCOFGMtI391R3NZHjHx3ZeFvLs4YZNR1u5GLTTbcbpHJ7nH3V9+vBxWX4Y8D313qcXijxtIt7rnWC0GDb2C9gg6Fh6849yAxAM620bWfibqEeo+JYJtO8LxsHtNHZislyR/HNjoPQfy6n023t4raBIbeJIoY1CIiKAFA6AAdAPSpRwPb/PFcp4m+IeheG5fsjStfao5KR6dZDzZmb0Kj7v40AdbmuJ1/4maLo96dMsUn1nWD8osdOTzGU9tzDhffqfasWHRPHXjrMniW8bw7oz8rpunv8A6RIvYSSdhjqPzAruNA8MaN4XsfsmjWENrH/EVGXf3ZjyfxoA4ePwf4r8cTG48b3xsNKblND0+TAI9JZB976DI7jaa77RtC0vw/YLZaTYQ2duv8ES4zjuT1Y+5JPvWlRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFF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proof	Plane Geometry	Math	Chinese
59	"如图, $\triangle A B C$ 的内心为 $I, D, E, F$ 分别是边 $B C, C A, A B$ 的中点. 证明: 直线 $D I$平分 $\triangle D E F$ 的周长.
<image_1>"	['如图, 不妨设 $A B \\geq A C, \\triangle A B C$ 的内切圆切 $B C, C A, A B$ 于 $T, K_{1}, K_{2}$, 过 $T$ 作内切圆的直径 $T K$, 过 $K$ 作 $\\odot I$ 的切线分别交 $A C, A B$ 于 $M, N$, 则 $N M \\| B C$.\n\n由于 $\\odot I$ 是 $\\triangle A M N$ 的旁切圆, $A K_{1}=A K_{2}$, 因 $M K=M K_{1}, N K=N K_{2}$, 所以有 $A M+M K=$ $A N+N K$.\n\n延长 $A K$ 交 $B C$ 于 $G$, 则由三角形内切圆性质知: $B G=C T$, 因此 $D T=D G$, 故 $D I$ 是 $\\triangle T G K$ 的中位线, 所以 $D P \\| A G$, 因 $B D E F$ 为平行四边形, 所以 $\\triangle D E P \\sim \\triangle A B G$, 相似比为 $\\frac{D E}{A B}=\\frac{1}{2}$.\n\n同理 $\\triangle D F P \\sim \\triangle A C G$, 相似比为 $\\frac{D F}{A C}=\\frac{1}{2}$.\n\n又注意 $\\triangle A M K \\sim \\triangle A C G, \\triangle A N K \\sim \\triangle A B G$, 相似比均为 $\\frac{A K}{A G}$.\n\n既然有 $A M+M K=A N+N K$, 所以 $A C+C G=A B+B G$, 因此, $D F+F P=D E+E P$, 即所证结论成立.']			['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proof	Plane Geometry	Math	Chinese
60	"如图, 四棱雉 $S-A B C D$ 中, $S D \perp$ 底面 $A B C D, A B \| D C, A D \perp D C, A B=A D=1, D C=S D=2, E$为棱 $S B$ 上的一点, 平面 $E D C \perp$ 平面 $S B C$.

<image_1>
证明: $S E=2 E B$;"	['以 $D$ 为坐标原点, 射线 $D A, D C, D S$ 分别为 $x$ 轴, $y$ 轴, $z$ 轴建立直角坐标系 $D x y z$, 如下图.\n\n<img_4121>\n\n设 $A=(1,0,0)$ ，则 $B(1,1,0), C(0,2,0), S(0,0,2)$.\n\n$\\overrightarrow{S C}=(0,2,-2), \\overrightarrow{B C}=(-1,1,0)$.\n\n设平面 $S B C$ 的法向量为 $n=(a, b, c)$, 由 $n \\perp \\overrightarrow{S C}, n \\perp \\overrightarrow{B C}$ 得到 $n \\cdot \\overrightarrow{S C}=0, n \\cdot \\overrightarrow{B C}=0$, 故 $b-c=0,-a+b=0$.取 $a=b=c=1$, 则 $n=(1,1,1)$, 又设 $\\overrightarrow{S E}=\\lambda \\overrightarrow{E B}(\\lambda>0)$, 则\n\n$$\nE\\left(\\frac{\\lambda}{1+\\lambda}, \\frac{\\lambda}{1+\\lambda}, \\frac{2}{1+\\lambda}\\right), \\quad \\overrightarrow{D E}=\\left(\\frac{\\lambda}{1+\\lambda}, \\frac{\\lambda}{1+\\lambda}, \\frac{2}{1+\\lambda}\\right), \\quad \\overrightarrow{D C}=(0,2,0)\n$$\n\n设平面 $C D E$ 的法向量为 $m=(x, y, z)$, 由 $m \\perp \\overrightarrow{D E}, m \\perp \\overrightarrow{D C}$, 得到 $m \\cdot \\overrightarrow{D E}=0, m \\cdot \\overrightarrow{D C}=0$, 故\n\n$$\n\\frac{\\lambda x}{1+\\lambda}+\\frac{\\lambda y}{1+\\lambda}+\\frac{2 z}{1+\\lambda}=0, \\quad 2 y=0\n$$\n\n令 $x=2$, 则 $m=(2,0,-\\lambda)$, 由平面 $D E C \\perp$ 平面 $S B C$, 得到 $m \\perp n$, 所以\n\n$$\nm \\cdot n=0, \\quad 2-\\lambda=0, \\quad \\lambda=2\n$$\n\n故 $S E=2 E B $ .']			['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']	Multimodal	Competition	False				Theorem proof	Solid Geometry	Math	Chinese
61	"如图, $F_{1}, F_{2}$ 是双曲线 $x^{2}-\frac{y^{2}}{4}=1$ 的两个焦点, 一条直线与双曲线的右支相切, 且分别交两条浙近线于 $A, B$, 又设 $O$ 为坐标原点.

<image_1>
求证:$|O A| \cdot|O B|=\left|O F_{1}\right|^{2}$."	['若直线 $A B$ 的斜率不存在, 即切点位于实轴的顶点, 则 $A, B$ 的坐标分别为 $(1,2),(1,-2)$, 这时 $|O A|=|O B|=\\sqrt{5}=\\left|O F_{1}\\right|$, 结论成立.\n\n若直线 $A B$ 的斜率存在, 可设直线 $A B$ 的方程为 $y=k x+b$, 由于 $A B$ 与双曲线相切, 所以关于 $x$ 的方程\n\n$$\nx^{2}-\\frac{1}{4}(k x+b)^{2}=1\n$$\n\n有两个相等的实根，即\n\n$$\n\\Delta=\\left(\\frac{1}{2} k b\\right)^{2}-4 \\cdot\\left(1-\\frac{1}{4} k^{2}\\right) \\cdot\\left(-\\frac{1}{4} b^{2}-1\\right)=0\n$$\n\n整理得 $k^{2}=b^{2}+4$, 由于 $A, B$ 的横坐标 $x_{1}, x_{2}$ 是方程 $x^{2}-\\frac{1}{4}(k x+b)^{2}=0$ 的两个实根, 我们有\n\n$$\nx_{1} x_{2}=\\frac{-\\frac{b^{2}}{4}}{1-\\frac{k^{2}}{4}}=1\n$$\n\n注意 $A, B$ 的坐标分别为 $\\left(x_{1}, 2 x_{1}\\right),\\left(x_{2},-2 x_{2}\\right)$, 可知 $|O A|=\\sqrt{5}\\left|x_{1}\\right|,|O B|=\\sqrt{5}\\left|x_{2}\\right|$, 因此\n\n$$\n|O A| \\cdot|O B|=5\\left|x_{1} x_{2}\\right|=5=\\left|O F_{1}\\right|^{2}\n$$']			['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proof	Conic Sections	Math	Chinese
62	"如图, $F_{1}, F_{2}$ 是双曲线 $x^{2}-\frac{y^{2}}{4}=1$ 的两个焦点, 一条直线与双曲线的右支相切, 且分别交两条浙近线于 $A, B$, 又设 $O$ 为坐标原点.

<image_1>
求证:$F_{1}, F_{2}, A, B$ 四点在一个圆上."	['在 $\\triangle O A F_{1}$ 与 $\\triangle O F_{1} B$ 中, $\\angle A O F_{1}=\\angle F_{1} O B$, 且 $\\frac{|O A|}{\\left|O F_{1}\\right|}=\\frac{\\left|O F_{1}\\right|}{|O B|}$, 所以 $\\triangle O A F_{1} \\sim \\triangle O F_{1} B$.\n\n同理 $\\triangle O A F_{2} \\sim \\triangle O F_{2} B$, 这样, 我们有\n\n$$\n\\begin{aligned}\n\\angle F_{1} A F_{2}+\\angle F_{1} B F_{2} & =\\left(\\angle F_{1} A O+\\angle O A F_{2}\\right)+\\left(\\angle F_{1} B O+\\angle O B F_{2}\\right) \\\\\n& =\\left(\\angle B F_{1} O+\\angle O F_{2} B\\right)+\\left(\\angle A F_{1} O+\\angle O F_{2} A\\right) \\\\\n& =\\left(\\angle B F_{1} O+\\angle A F_{1} O\\right)+\\left(\\angle O F_{2} B+\\angle O F_{2} A\\right) \\\\\n& =\\angle B F_{1} A+\\angle B F_{2} A\n\\end{aligned}\n$$\n\n即四边形 $F_{1} A F_{2} B$ 中一组对角之和等于另一组对角之和, 从而对角之和为 $180^{\\circ}$, 该四边形内接于圆.']			['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proof	Conic Sections	Math	Chinese
63	"如图, $D$ 为锐角 $\triangle A B C$ 内一点, 使得 $\angle A D B=\angle A C B+90^{\circ}$, 且 $A C \cdot B D=A D \cdot B C$, 延长 $A D 、 B D$ 、 $C D$, 分别与 $\triangle A B C$ 的外接圆 $\Gamma$ 交于点 $G 、 E 、 F$.

<image_1>
证明:  $E F=F G ;$"	['注意到,\n\n$$\n\\begin{gathered}\n\\triangle E F D \\sim \\triangle C B D \\Rightarrow \\frac{E F}{F D}=\\frac{B C}{B D}, \\quad \\triangle F D G \\sim \\triangle A D C \\Rightarrow \\frac{F G}{F D}=\\frac{A C}{A D} \\\\\nA C \\cdot B D=A D \\cdot B C \\Rightarrow \\frac{B C}{B D}=\\frac{A C}{A D}\n\\end{gathered}\n$$\n\n故\n\n$$\n\\frac{E F}{F D}=\\frac{F G}{F D} \\Rightarrow E F=F G\n$$']			['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proof	Geometry	Math	Chinese
64	"如图, $D$ 为锐角 $\triangle A B C$ 内一点, 使得 $\angle A D B=\angle A C B+90^{\circ}$, 且 $A C \cdot B D=A D \cdot B C$, 延长 $A D 、 B D$ 、 $C D$, 分别与 $\triangle A B C$ 的外接圆 $\Gamma$ 交于点 $G 、 E 、 F$.

<image_1>
证明: $\frac{S_{\triangle E F G}}{S_{\text {圆 } \Gamma}}=\frac{1}{\pi}$."	['由\n\n$\\angle E F G=\\angle E F C+\\angle C F G=\\angle E B C+\\angle C A G=\\angle A D B-\\angle A C B=90^{\\circ} \\Rightarrow \\triangle E F G$ 为等腰直角三角形\n\n于是, $E G$ 为圆 $\\Gamma$ 的直径, 设圆 $\\Gamma$ 的半径为 $R$, 故 $\\frac{S_{\\triangle E F G}}{S_{\\text {圆 } \\Gamma}}=\\frac{\\frac{1}{4}(2 R)^{2}}{\\pi R^{2}}=\\frac{1}{\\pi}$.']			['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proof	Geometry	Math	Chinese
65	"<image_1>

如图,已知锐角 $\triangle A B C$ 满足 $A B>A C, O 、 H$ 分别为 $\triangle A B C$ 的外心、垂心,直线 $B H$ 与 $A C$ 交于点 $B_{1}$,直线 $C H$ 与 $A B$ 交于点 $C_{1}$. 若 $O H / / B_{1} C_{1}$, 证明: $\cos 2 B+\cos 2 C+1$ $=0$."	['联结 $A O 、 A H$.\n\n因为 $A 、 B_{1} 、 H 、 C_{1}$ 四点共圆, 且 $O H / / B_{1} C_{1}$, 所以, $\\angle O H C_{1}=\\angle H C_{1} B_{1}=\\angle H A B_{1}$ $=90^{\\circ}-\\angle A C B$.\n\n又因为 $B 、 C 、 B_{1} 、 C_{1}$ 四点共圆, 所以,\n\n$\\angle A H C_{1}=\\angle A B_{1} C_{1}=\\angle A B C$\n\n故 $\\angle A H O=\\angle O H C_{1}+\\angle A H C_{1}=90^{\\circ}-\\angle A C B+\\angle A B C$\n\n由 $\\angle O A B=\\angle C A H=90^{\\circ}-\\angle A C B$, 知\n\n$\\angle O A H=\\angle B A C-\\angle O A B-\\angle C A H=\\angle B A C-2\\left(90^{\\circ}-\\angle A C B\\right)$\n\n$=\\angle A C B-\\angle A B C$.\n\n则 $\\angle A H O+\\angle O A H=90^{\\circ}$.\n\n从而, $\\angle A O H=90^{\\circ}$.\n\n设 $\\triangle A B C$ 的外接圆半径为 $R$. 则\n\n$A H=2 R \\cos A$.\n\n由 $A O=A H \\cos \\angle O A H$, 知\n\n$R=2 R \\cos A \\cdot \\cos (C-B)$\n\n$\\Rightarrow 2 \\cos A \\cdot \\cos (C-B)=1$\n\n$\\Rightarrow \\cos (A+C-B)+\\cos (A-C+B)=1$\n\n$\\Rightarrow \\cos 2 B+\\cos 2 C+1=0$.']			['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']	Multimodal	Competition	False				Theorem proof	Geometry	Math	Chinese
66	"<image_1>

如图,已知锐角 $\triangle A B C$ 的外接圆为 $\odot O$, 过点 $A$ 作 $\odot O$ 的切线 $l, l$ 与直线 $B C$ 交于点 $D, E$ 为 $D A$ 延长线上一点, $F$ 为劣弧 $\overparen{B C}$ 上一点, 直线 $E F$ 与劣弧 $\overparen{A B}$ 交于点 $G$, 直线 $F B 、 G C$ 分别与 $l$ 交于点 $P 、 Q$. 证明: $A D=A E$ 的充分必要条件为 $A P=A Q$."	['设 $E F$ 与 $B C$ 交于点 $H$.\n\n对直线 $G C Q$ 和 $\\triangle H E D$, 应用梅涅劳斯定理得\n\n$\\frac{H G}{G E} \\cdot \\frac{E Q}{Q D} \\cdot \\frac{D C}{C H}=1 \\quad \\quad (1)$.\n\n对直线 $P B F$ 和 $\\triangle H E D$, 应用梅涅劳斯定理得 $\\frac{H F}{F E} \\cdot \\frac{E P}{P D} \\cdot \\frac{D B}{B H}=1 \\quad \\quad (2)$.\n\n(1) $\\times$ (2) 得\n\n$\\frac{H G}{G E} \\cdot \\frac{E Q}{Q D} \\cdot \\frac{D C}{C H} \\cdot \\frac{H F}{F E} \\cdot \\frac{E P}{P D} \\cdot \\frac{D B}{B H}=1 \\quad \\quad (3)$.\n\n由相交弦定理和切割线定理得\n\n$H B \\cdot H C=H F \\cdot H G$,\n\n$E G \\cdot E F=E A^{2}$,\n\n$D B \\cdot D C=D A^{2}$.\n\n代人式(3)得 $\\frac{D A^{2}}{E A^{2}} \\cdot \\frac{E Q}{Q D} \\cdot \\frac{E P}{P D}=1$.\n\n故 $A D^{2}(A Q+A E)(A P-A E)=A E^{2}(A Q-A D)(A P+A D)$.\n\n整理得\n\n$(A D+A E)[A P \\cdot A Q(A D-A E)+A D \\cdot A E(A P-A Q)]=0$\n\n$\\Rightarrow A P \\cdot A Q(A D-A E)=A D \\cdot A E(A Q-A P)$.\n\n因此, $A D=A E$ 的充分必要条件为 $A P=A Q$.']			['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proof	Geometry	Math	Chinese
67	"如图, 已知 $\triangle A B C$ 满足 $A B$ $<A C, M$ 为边 $B C$ 的中点, 过点 $A$ 的 $\odot O$ 与边 $B C$ 切于点 $B$, 且与线段 $A M$ 交于点 $D$, 与 $C A$的延长线交于点 $E$, 过点 $C$ 且平行于 $B E$ 的直线与 $B D$ 的延长线交于点 $F, F E 、 C B$ 的延长线交于点 $G$. 证明: $G A=G D$.

<image_1>"	['因为 $\\angle D A C=\\angle E B D=\\angle D F C$, 所以, $A 、 D 、 C 、 F$ 四点共圆, 设该圆为 $\\odot O_{1}$.\n\n由 $M C^{2}=M B^{2}=M A \\cdot M D$, 知 $M C$ 与 $\\odot O_{1}$切于点 $C$.\n\n因为 $B E / / C F$, 所以, $G$ 是 $\\odot O$ 与 $\\odot O_{1}$ 的一个位似中心.\n\n又 $A D$ 是 $\\odot O$ 与 $\\odot O_{1}$ 的公共弦, 于是, 点$G$ 在 $A D$ 的中垂线 $O O_{1}$ 上.\n\n故 $G A=G D$.']			['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']	Multimodal	Competition	False				Theorem proof	Geometry	Math	Chinese
68	"<image_1>

如图,已知锐角 $\triangle A B C$ 的外接圆为 $\odot O, A D$ 为 $\odot O$ 的直径,过点 $B 、 C$ 且垂直于 $B C$ 的直线与 $C A 、 B A$ 的延长线分别交于点 $E 、 F$. 证明: $\angle A D F=\angle B E D$."	['<img_4146>\n\n如图, 联结 $B D 、 C D$.\n\n则 $\\angle D B F=\\angle D C E=90^{\\circ}$,\n\n且 $\\frac{B F}{C E}=\\frac{\\frac{B C}{\\cos \\angle C B F}}{\\frac{B C}{\\cos \\angle B C E}}=\\frac{\\sin \\angle B C D}{\\sin \\angle C B D}=\\frac{B D}{C D} .$\n\n故 $\\triangle D B F \\backsim \\triangle D C E$\n\n$\\Rightarrow \\angle B D F=\\angle C D E$\n\n$\\Rightarrow \\angle B D E=\\angle C D F$.\n\n因为 $B E / / C F$, 所以, $A$ 是线段 $C F$ 与 $E B$的一个位似中心. 设点 $D$ 关于此位似变换的对应点为 $D^{\\prime}$. 则\n\n$\\angle B D E=\\angle F D C=\\angle B D^{\\prime} E$.\n\n于是, $B 、 D 、 D^{\\prime} 、 E$ 四点共圆.\n\n故 $\\angle A D F=\\angle A D^{\\prime} B=\\angle B D^{\\prime} D=\\angle B E D$.']			['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']	Multimodal	Competition	False				Theorem proof	Geometry	Math	Chinese
69	"如图, 已知锐角 $\triangle A B C$ 的外接圆为 $\odot O$, 其垂心、内心分别为 $H 、 I, A H$ 的中点为 $M$, 且满足 $A O / / M I$. 设 $A H$ 的延长线与 $\odot O$ 交于点 $D$, 直线 $A O 、 O D$ 与 $B C$ 分别交于点 $P 、 Q$.
<image_1>
证明: $\triangle O P Q$ 为等腰三角形;"	['显然, $\\angle O A D=\\angle O D A$.\n\n因为 $A D \\perp B C$, 所以,\n\n$\\angle O A D+\\angle A P C=90^{\\circ}$,\n\n$\\angle O D A+\\angle D Q C=90^{\\circ}$.\n\n又 $\\angle D Q C=\\angle P Q O$, 则\n\n$\\angle O D A+\\angle P Q O=90^{\\circ}$.\n\n于是, $\\angle P Q O=\\angle A P C=\\angle O P Q$, 即 $O P$ $=O Q, \\triangle O P Q$ 为等腰三角形.']			['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proof	Geometry	Math	Chinese
70	"如图, 已知锐角 $\triangle A B C$ 的外接圆为 $\odot O$, 其垂心、内心分别为 $H 、 I, A H$ 的中点为 $M$, 且满足 $A O / / M I$. 设 $A H$ 的延长线与 $\odot O$ 交于点 $D$, 直线 $A O 、 O D$ 与 $B C$ 分别交于点 $P 、 Q$.
<image_1>
证明: 若$I Q \perp B C$，则$\cos B + \cos C = 1$."	['如下图, 由 $I Q \\perp B C$, 知 $Q$ 为 $\\triangle A B C$的内切圆 $\\odot I$ 与 $B C$ 的切点.\n\n<img_4093>\n\n则 $B Q=\\frac{B A+B C-A C}{2}, N$ 为 $P Q$ 的中点.故 $B P=2 B N-B Q=\\frac{B A+B C+A C}{2}-B A$.\n\n因此, $P$ 为 $\\triangle A B C$ 中 $\\angle A$ 内的旁切圆 $\\odot I_{a}$ 与 $B C$ 的切点.\n\n设 $P_{1} Q$ 为 $\\odot I$ 的直径, 过点 $P_{1}$ 作 $B C$ 的平行线, 与 $A B 、 A C$ 分别交于点 $B_{1} 、 C_{1}$. 则 $A$ 为 $\\triangle A B_{1} C_{1}$ 与 $\\triangle A B C$ 的位似中心.\n\n因为 $P_{1} 、 P$ 为对应点 $\\left(P_{1}\\right.$ 为 $\\triangle A B_{1} C_{1}$ 中 $\\angle A$ 内的旁切圆 $\\odot I$ 与 $B_{1} C_{1}$ 的切点, $P$ 为 $\\triangle A B C$ 中 $\\angle A$ 内的旁切圆 $\\odot I_{a}$ 与 $B C$ 的切点), 所以, $A 、 P_{1} 、 P$ 三点共线.\n\n又 $O N$ 为 $Rt \\triangle P P_{1} Q$ 的中位线, 则\n\n$O N=\\frac{1}{2} P_{1} Q=I Q$.\n\n设 $\\odot O, \\odot I$ 的半径分别为 $R$、$r$. 则在 $\\triangle C O N$中,有\n\n$R \\cos A=r \\Rightarrow \\frac{r}{R}=\\cos A$.\n\n由 $\\cos A+\\cos B+\\cos C=1+\\frac{r}{R}$\n\n$\\Rightarrow \\cos A+\\cos B+\\cos C=1+\\cos A$\n\n$\\Rightarrow \\cos B+\\cos C=1$.']			['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAcsByAMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACikJApNwoAdRSFgKyL7xb4c0y7ktL/X9KtbqPG+Ge8jR1yMjKlsjgg/Q0AbFGa85vvjl4DtLWSWHU572VcYt7e0kDvk9i4VenPJHA+grnbv9o/w4pX7Lo2qyDnd5wjjx6YwzZoA9nBzS14f/AMNJ6QP+Zfvv+/qUv/DSekY/5F++/wC/yUAe30ma8RP7SekD/mX77/v8lJ/w0npBH/Iv33/f5KAPcKK8ei/aN8JmNfM0vWlcjlVhiIB9j5grobD41eAb77Mv9t/Z5Zyo8u4t5E8tm7O+3YME8ndgdc45oA9AorK07xNoWr3DW+mazp97Mil2jtrpJGCggZIUkgZI5rU3fWgBaKTcBS0AFFFFABRRRQAUUUUAGeaTNIxx6cAnmuU8JePdN8Y6rrdlpqS7NKmWIzkqUmyWAZCCcglGx7EH6AHW0UgORS0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRSFgDzQAtITijPNeW+NPjh4a0CN4NJlXWNRA+UW7ZgQkDBaTowwx4TdypBK0Aeo7gf/rVia74y8O+GcDWdXtbOQrvELvmQrz8wQZYj5TzjtjrXhqeNfih8UmFn4dtf7MsA4WW6td0argoTunY5yuQdseGKseGFbelfs5W7lZ9d8RXE8kgDSpaRhcORlvnfduG49Soz6c8AGj4g/aH8O2Ehi0bT7vVXVlHmMfs8TKVySpIL5BwMFB3OeBnj5vHHxl8Sok+m6Xc2VpJApH2XTtscinkOryhicgjlWxgA+59l0D4aeE/DdvCllotpJNEyyLdXUYmm8wAfPvYfKcjOFwAeQBXWbTnqKAPnqP4U/FLxJaouv8Aicw2txIPtNpcX8spRQ/Xy1zGx4DABh25B6bWm/s26NDHINU16/unLAo1tGkAA9w2/P5ivbFGBiloA4Oz+DXgOyIZdBSV9mxmnnkk3e5BbGeOwq4PhX4HBz/wjVjnOfun/GuwooA54eA/CAGP+EV0P8dPiP8A7LTl8D+FI2LR+GNEQ4I40+IZBGCPu+mR+JrfooA5/wD4QTwiSS3hbRGJOSTp8XP/AI7R/wAIJ4Q/6FXQ/wDwXRf/ABNdBRQByD/C7wRJIzt4a08FjkhUKj8ADgViar8CvBGpREQ2Vxp8hk3mW0uWyRg/Lh9ygc9AB0GD1z6VRQB4Xrv7N9i9qG8P61cRXCq2U1ALIkh4wNyKpQdcna3bgd8a98I/G3QpmW01m81KMxh2lh1LzQuCflAmKtnvwMHI619G0mKAPniD4ofE7weqyeK/D8l5ZF9zTzW5gIyMKgkQbF5GcFSTz6jHY+G/j54T1O1J1fz9HugeUkRpo2ySPldFzwACdyrjOBnGa9UKk9+1cT4l+EnhDxRLNc3WnC2vpcFruybynLbtzMRyrMSSCzKTz14GADptJ8R6LryM+k6rZ3wUKXFvMrlNwJAYA5UnB4ODwfQ1pbhXz/qvwC1jQ9up+EPEcpv4QSqOTBIflYHZKp+8eFwcDk5YdKqRfFnx74CFppni3QPPWJTGs1yGSWfbjnzhuSTAIGQDnIySckgH0WDnpS15/wCDfi/4Y8WiK2+0HTdTfC/ZLwhdzHaMRv8AdfLNgDhjgnaK74sB1zQA6kJAoBz2IpGPOKAOI+KPiZtF8LSWFhcSJruq/wCi6bFESJJJGZVbaQDtID5zxzjBzivJf2cdQmj8U6zparGYbiwFwzMCWDRuFUdemJWyPYdKNQ8Wa9q/xZk8Saf4f1LxHoujzy21jFaRl4A23aXDojKSSd+evKc4Aqh+zzcRRfEW6SaZFebTJY4lZxl28yNiBnqcKx+gJoA+oQMD8aWkBzS0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFISB1oJxXL+M/Hmg+CbIXGq3JMz7RHZwbWnkBJGQpI+UYPJIHGM5IBAOn3DGSCPwryzxh8dPDmgtPaaVnV79VGxoHX7OCcHmQHng/wg8jBK9R55car8RPjTLJp1paiw0KSRpdxRktwF+6ry7SZCDt+UDkndtAX5fUfC3wV8J+HHgupbd9T1CHB8+7PyB8YJWMfKBk5GdxGBg5GaAPL7rwj8WPiY32nVibCzkJ8u3u3NvEgDMQPJUFhgjALqWI28kYNep+D/AIOeGPC8SSz2yatqGMNdXiAhfu/cj5VcbeDywyRuwcV6Ft6d8etOFAESQrFGqRqqKq7VVRgAdgAOlSAYzS0UAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQA1lJ71HPbRXMLRTxxyxt1R13Ke/Q1NRQB5L43+BWia+Gu9C8nR78Lny0j/wBGlIXABQfcyduWX3JVic1wsU/xM+DbJdagf7R0HzEheN7rzYurHCZ+eI43cgYyVyDwK+kyKintYbqCSC4ijlhlQpJHIoZXUjBDA9R1496AOE8GfGDwx4taK085tO1J1X/RbshQ7kgbI3zhzuIAHDHrt64h+MWseI7LwzHp/hnTb+6utQLRzTWto03kwgYYZGdrNuABweA54IBFHxh8DPDuvme60rOkXzINiwIPsxIwOY8ccA8qRyc4PfhdG+I3jr4bagmn+ONPv77T2wiPMwZ1JIZmSbkS4VvulzjgZXBFAHq/gKWHSPhFo9zDp946Q6cJjbRxKZZDgs2xRgNuOSOeQQTya8V8Pxa5pXxim8X3Hg7xM2nve3VwI10x/NCyiQLkHAz84z834nv9D+HPE+leKtIg1HSrtJY3VWaPcu+JiM7HAJ2sPT+fWtjt34oAVDladSDpS0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRSFgOvegAJxRuHPt6c1T1TU7LR9Om1DUbmO2s4F3SSyNgKP8ScDHfOOpr5+8XeOPE3xR1q58P+Bre5bSoonWVkcRNcocZZ2YgKh6BTgkE567QAdB41+N5kvrjw/wCCbSW/1BiIor+ICRd4IJ8pNreZwCM8DIyNw61fB/wElmvW1fxxdC7km3SPZRzOWZ2GSZZQQSwJOQpOSAdxGQe98A/DHRvB+mWUr2dvPrccZ869ILHe33tm7ooB2ggAkdeprugMDmgCOOFYo1jRVREACogwFx6Y6VIBgYpaKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKQnFBOKz9Z1zTfD+mSalqt0ltax4BdgTyegAGSSfQUAWL27gsLSe8uZViggiaWWRuiIoySfYAGvlD4pfE4+P7q2t7Sze10y0LGLzSDLI5/ibHAAAwFBOOSScgC18U/iRf8AjK/urfSZ7pfC8DpEAqlFnflg79+SpKq3ZAcA5w34feCRqHhDxb4i1TTS1pb6TP8AYZJVIV5QrHenrtKYz6nvzQBxfhz+24NQm1XQPPW70uI3jyQEboowQrMR/EvzgMMEbScjGa+kPAHxp0fxU9vpuprHpery7URCT5VxIc52N/Dkj7rHOWABY14X8Jr5NP8AinoE0mNrTtBycYMiMg/VhXtHxG+Cum65Yyah4YtoLDVIkGLSFVihuFGcrgABG9G6HGD13AA9e3ClBzXgXgL4r6noGsXXhn4hzSwNGcR3Vyh8yNuPkfAO5SDuDn8yCNvvFvcwXFtHcW0qTQSqHjkjYMrqRkFSOCCPSgCaigHNFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFITigAJxWP4m8R6b4W0G51jVJWS2gXoo3M7H7qKOhYn16dTgAkS67r+k+HdNfUNYvobS0U7d0p+82D8qgcscAnABOBXz5Zr4m+Ofixjey3dp4Tt52b92uEiAHCjs0pB5JzjeSBjC0ARz3ni/47a/a28dq+n+HraXLMoYwwkY3FnwPMlwwAUYwD0UFmr6B8MeGNL8JaHDpOkW/lW8fLMx+eVyOXc92P8gAMAACfRdD03w/pcWnaTaJa2cXKRqSee5JOST7kmtIDAxQAAYpaKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKQnFBbBrJ8R+IdM8MaPNqmqTr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iiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKQsF60E4ryj4r/ABWTw5CujeHLhJ9flYB9iiQWw6YI5HmE8BME4yTjjIBo/Ev4q6b4KtprC2LXGvSQboYVHyw7shXc+3J2jJOOcA5rk/hX8K9QfU5PFXja3NxcyqHtre9cyybz1klB/iAwAGyRkkgECpPh18I7m9li8VeOpLq61SRkmhtbmVmZduNrTE8luBhM4AGGzkqvtwGKAGwxrDEsaIiIgCoiDAUDgAU+ikLAYz3oARjj09s+tfPPxT1XUPH3xKsvAejXKC2gk2SlJso0mN0jON2D5aqeOoIYck4r0n4ueNJvBvg9rjT7qKHVLmRYrTcAzDnLuFIOQBxzwCwz1AON8DfBSaH4U/tm7tJINU1HcD5ybXjiDEKACoKg43eh+U+mAD0rStJstE0y303TbZLWyt0CRRRjhR1PXqSeSTySSScmp7qeG0tpJ7iVIoYkMkkkjBVVRySSegA7niphxXEfFjXbPRvh9rEU97bw3N5aSQwQyON827CsFXqeH5x0zk4oA8D8SXQ1zwN4g8UhpXTUvFaLEZ3JkWNIZWRTyQMLIAACcYx0ArX8I3C/EDw54W+HFv8Aara3s5J7zVrhYxkoHZkEZ3dzJg7l4baQDjBt+Ko7Kf8AZt8OLo0z3cNnexteP9428rJKZFfgbQHkAGezJycgl+naZp/w3+DGoalqscf/AAkHiW0e2t7eTCyLE424GMnAVhI3TnYpwcUAYHww0XTte+NIk063zo9lcTX0MbuylYkY+TyTkkM0ZwTzg5z0r6X8Rx2MnhfVYtSmeDT2spluZYwS0cRQ7mGATkDJ6V4p+zkNIjbWZZL22/teZkjS2dAsiwqCxZWPUMTyB08tSeoro/j94rm0bwtb6LZ3Hlz6o7Cba3zeQuNw+jEgHPUBhyCcAC/s5/8AJPb/AB/0FZP/AEVFXr1eQ/s6c/D6/wD+wrJ/6Kir16gDyX9oTS0vPAEV/wCUplsbtG8zy8sqNlSN38IJKZ9SorsPhvri+Ifh7ot95ssswtlhuGmbdIZY/kcscknLKTk8kEE9an8eaEfEvgnWNJSNpZ5rVjBGrhS0q4eMZPAG9Vznj6V5/wDs66tDceCb7TPNBuLO9LlAhG2ORQVJPQ5ZZO+ePpQB7HRRRQAUUUUARzIkqNHIodGUqysMgg+1fOfgeS5+GfxouPDeoyTQ6bqDmG3XDmKTew8lwM9SQE3fNg7hngkfR5Ga8e/aA8L2974STxAkY+26fIiPIAMvC7Bdp9cMVI64y3HJNAHsK9KWuR+GvipvF3gXT9TnZWvQvkXWCP8AWpwWOAANww2AMDdjtXXUAFFFFABRRRQAUUUUANY849uK+dPG6D4o/G2z8PWBmNjp4+z3cqgjYEYmZgdpwf4ASNpbHY5PtPj7W5fDngXWdVgkEc8FsfJcgHbIxCoeQQcMw4I56V5T+zjoifY9a12SNS7yJZxPjlQAHcDB77o+38IoA92hRIoljiVVjQBVVRgAdgPapKRelLQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFISBQTg4wa8Y+J3xVuk1FfCHgsvPrMkoinuIRkwtu/1Ser8fMeijI652gFz4pfGK38MI+k+H3hu9YdGDzpIrpZkEg7gM5cYPynGOCcjgx/Cb4SzeG7qPxNrs/m6tLDmK3AP+jFx8xYnkvg7T2HzfeyCJvhf8HbLw1bQ6tr1vFda2zLKkbfNHZkcgAdGcHktzggbehLesgYoAFBA5paKKACobqeG1t5J7maOGCNC8kkjBVVRySSeAAKlLAEA9T0rwr49eKbq8u7DwPpDRzTXTo11FGd0hcsPKiPYZPzY6/dPA6gHP+FrKf4xfFu88QX6H+xtPkSTyXwR5YJ8mEoxbhtpZwPlPz9Nwr6UX7tc/wCCfDVt4V8JWGlW9vFC6RK1yUHMk5A3uxyckkepwAAOAK6EDAoAWqWp6Rp2swLBqen2l9Arb1iuoVlUNgjIDAjOCefc+tXaKAKMOj6db6WdLgsLWLTijRm0jhVYirZ3DYOMHJyO+TRqOj6drEIg1OwtL2BXDrHcwrIoYAgHDA84JGfc+tXqQkCgDItNJ0Tw1BdT2Gm2Onxld85tLdYy6qCRnaBnGWx1xzXz3oOlzfG34ha5qmoStbWMNsywptZxDuVkhA+YAEHMhwcMwbjDHHefHTxwdK0aPwtprO2p6qg8wLGTttzuUgHP3nYbeh43dMqa6v4ZeCovBfhC2tpLdE1SdBLfyADcznkISCchAdowcEgtjLHIByv7OX/JPtQ/7C0n/oqKvX68g/Zy/wCSe6h/2FZP/RUVev0ANYc84xXz/wCEZE8C/tB6xokrCCx1RmW3SM+VCC5EsQweDgFoxju2B1xX0CRmvnX43W8uh/FTw74gCiG3kSJ/PyGzJDLlvlOeimPtg+/NAH0UOlLTVIIOPWnUAFFFFABVXULG31PT7mwu4/NtrmJ4Zo8kbkYEMMjkZBPIIq1SMM0AfPXwn1KX4ffEPU/AmtO3+lTKLeXzl8pZApKkKTgeapTp82Qiken0In3eK+ffjvpkug+MtA8YW6yvllV9qkKkkTB0y/IBZScDH/LMnnkV77ZXUF9ZQXdrKstvOiyRSKchlYZBH1BFAE9FFFABRRRQAUmcUtNbrQB4p+0R4nms9FsPD1rLHjUCZrtQQXEaFSnHUAtk545jwD1FeoeD/DUPhHwrYaHBcPcLaqwMzgKXZmLscDoMscDnAxyeteH6tPJ4l/aes7eQmWCwuI0hVgo2CFDKRlRzhw555PQ+lfRi9KAAUtFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFITigBaQnFJuFeBfEn4k33i7WE8EeB2e4S4fybi5tzg3B/iRG7RgAln6EZ6KDuAL3xR+LzGSPw34Julury6AjlvLMeYVL42pCyk5c8gkZxkAHdnb0Pws+E1p4OtYdW1FfO1+WHDbyCtpuHzKmON2Dgtk9DjAJzP8MvhRb+BVlvru7S+1edQrSqmEgGPmVD1OWzljjIA4HOfSAMDAoAFGB+NLRRQAUE4oprdetAHM+PvFcfg/wdfawBFJcIojtopDxLKxwoxkbgOWIHOFPpXl/wB8KSXct9431RpZrmaR4LZ7hdxYnBkm3MMsSfk3A/89AevGLqdxcfF74zx6XCyz+H9NkBkjjnPlyQxuBJJw2CzFtoZedpX0Jr6NtoIba2jt7eKOGCJQkcUahVRQMAADgADoKAJAMZ5zS0UUAFFFFABVXUb2303T7m/u5PLtbWJp5nwTtRRuY4HJ4Bq0TivGPj94kT+yLPwfZLPNqmoyxytDHGxzFuYKox1ZpFGAAfunp8uQDlfCdm/wAXfi/feI7oNHpOmypMiCNcsqt+4jYEn7wVmY8jgjjK4+jx0Ncd8MPCM/g3wRbaZeeT9tZ3mufKO5d7HjnAzhQoPXkdcYrscYB+lAHkX7OP/JPdQ/7Csn/oqKvX68g/Zx/5J7qH/YVk/wDRUVev0AFeUftAaN/aHw9XUFS3Emm3SStI4+cRv+7Kqcd2aMkcZ2+oFer1na7YjU9B1GwKhxdWskG0nAO5SuP1oAwPhZqY1X4Y+H7jyfJ2Wot9u7dnyiYs599mcds45xmuwrwH9m/VpSmu6O8h8lTHdRIFGFJyrknrziPjpwele+r09PagBaKKKACiiigDkfib4fj8SfD/AFazaMNNHCbm3baCVljG4YJ+7nBUn0Y1x37Pnie41jwneaReTzzzaXKgiaTGFgdfkQHqcFH69AVAOBgetzRrKjRuqsjqVKsMggjuO4r5y+BYfw/8Udb8PXk9v5qwywHa/EssUijCZxnje3TOAT2NAH0jRSL0paACiiigAqG5nhtoZJ7iRI4YkLySSEKqKBkkk8YxU1cF8YteXQPhrqjbo/Ovk+wwrIpIYyAhunQiPewJOMgdehAPNfgLYS6/408QeLb8RvcITjMOQZZmLM6tn5SApGB2kPbg/QqjArzX4EabDZfCyzuI2cvfXE1xIGwQrBzF8vHTEa+vJNeljigBaKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoopCcUALnmoppUijaSR1SNBudmbaFA7k9hSXNzDa28txcTRwwwoXkklYKqADJJJ4AA5zXzv408Zan8Xdfg8I+EI5/7NDlpnkJRZgG/wBZJ3WJeCAeSSPl3bQACz8RfiTq3i7WLjwP4OtzcRSyG3kubZyzXQwNyrwAqA7wzZIKjOQuc+k/DX4aWHgPSgzeVcazOgW7u1BIxnPlx55Cg4z03EZIHAXS8C+BdN8C6EtjZKJLqQBru7YfPPJ6+yjJ2r2z3JJPUgYFAAowMZpaKKACiijPNACFsHHP4V5F8ePGkuheH4dEsLmJLzUw63CfKzi2KlW4IO3cWwG4+62Oeno/iXX7Lwx4fvNZvy4t7aPcQi5ZmJCqoHqWIGTxzzxXgfwm0G5+IHj/AFHxrrwkZbS4W4jALhGuM5RQTn5IwB8u7I+TqMggHqHwh8FReE/B0M0ltLDq2oIkt75uQwxuKJsJ+XaHII4OSc9gPQlBAOfWkQYX0p1ABRRRQAUUUhYL1oAiubiG1gkuLiVIYIUMkkkjbVRQMkkngADJr5x+FGnXPjj4raj4s1CJ5rW1kedZZwGAmY4iXBJwVXJGMhdi9Plro/2hvGBs9NtvClsGWS8Vbq6fAx5Ssdi9Opdc5BGNg6hjXpXgLwtD4R8H2GlxoomVPMumwuXmblySOuD8ozk4VRk4oA6Yd6D3+lKKQ9/pQB5D+zj/AMk91D/sKyf+ioq9fryD9nH/AJJ7qH/YVk/9FRV6/QAU1hk8elOpCM0AfO/hyJPBH7SV7pUYt47LUTJGhaPykRJVEyJGM4GHVIx2OMAA4x9EKcjNeD/tB2jaZqvhnxNYbItQgdoxMeTlCJIsKcqQpL9ucjrXtei6kms6Fp+qxxtGl7bR3KoxyVDqGAP0zQBeooooAKKKKAEYE9DXzN8YE1HwZ8YbXxPZPJvuFiuoXKMibowI2iLAjcCqgsAQcSYPXJ+mq8S/aS08S+HdF1HC5gu2t85Of3iFvpj91/L1NAHta8g806ub8AahJqvw/wBBvZpJJJpLKISSSPvZ3ChWYnuSQSc8810lABRRRQAV4l+0jc7PD2iW3m4El28hjz97amM/hv8A1r22vn39paeFrnw1As0ZnjS5do9wLKrGIKSOuCVbB/2T6UAesfDS2e1+Gnh2NypJsY5Bt9GG4fjg/wD6+tdVVDRNNTRtC0/S43aSOyto7dXYYLBFCgn8qv0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRSE4oACcVU1HU7LSdPmv9QuorW1hGZJpmCqvOByfUkADvkVJd3dvZ2k95cyrFbwRtJK7dFUDJJ+gFfOWqa7rfx28S2+g6XANP0W0ka4eWU7mVPuiRwDgtg/Ki92IyQNwAJ/FPjPxN8WNafw34Qtpk0MS+TNcANtlBJxJK+Mxx4UkL1ODnccAeyeBvAml+BdGFlp4824lwbu7cYedxnBx/CBkgKOgPUnJNjwj4P0rwdosVhpcEYcKBPclAJZ2BJy7AZbBY4B6A4FdCowKAADFBIHU0E4rzj4r/ABNHgWyt7SxiS41e9VjErOMQION7KOTk8KOAcNz8uCAd/dahZ2Pki7uYbczyiGHzZAnmSHOEXPVjg4A9Kn3CvE9K+C8vi3T4dc8da7q0+rXcYl8qNlT7OrEt5eHQ4wWOVAUKcgA9azfh74nv/BvxMuPh7dXr3+km4eC2knJDwMF3Lt6/K2ANvTJyMchgD38HIzSMf/rfWhSMVw3xV8bnwX4RkuLWSMandEwWisRuBxzIFIOduR1GMkA9RkA8t+LOuXvj3x/ZeB9FdZba2n2yCMEkz87y2SB8ihuB/tc+nvPh/QrPw5oVnpGnqy2tpH5abjlm5JLHtkkknAHJNeZfAfwPFpHh5fFFyko1LUoykavlRHb7hjCnruKht3PG3GOc+wKMDmgBQMUUUUAFFFFABWdreq2+haPd6reFvs9pC0rhBlmwM4UEgEnoK0CwBANeH/HTxXcX91ZfD/RwJbu+kiN0oZOSzDyosk/KSwVjnHG3nBNAGB8HNEv/ABt46u/Guvv9q+xMCJXRcS3G0BeAu3CKM8YwfLIr6PUYFc/4M8J2ngzwxa6NZt5nlZeacxhGmkJyzED8AM5IUAZOM10AoAWkPf6UtIe/0oA8h/Zx/wCSe6h/2FZP/RUVev15B+zj/wAk91D/ALCsn/oqKvX6ACiiigDzf44aAmtfDe8nSGSS601lu4vLAztB2yZyPu7GZjjuoPao/gRfC5+FtnDuQ/ZbiaH5TkjLl+ff5/yxXoWoWNtqen3NheR+ZbXULwypkjcjAqwyORwT0rxT4GXj6J4p8VeCJLiSdbOd5YXESqhMb+VIx53At+7wMkDB/EA91opqHK5FOoAKKKKACuD+Mmltqvwv1hYrWOee2RLqMsBmMI4Z3BPQiMP05IJA613lZ+t6aus6JqGlyO0cd5bSW7OoyVDqVyPpmgDkPgvqaal8LNJAkRpbXzLaVV/gKucA++0ofxrv68V/Zv1GF/C+s6UFfz7a9Fw7cbSsiBVAOeuYmzxjkde3tVABRRRQAZr51+Jd7a6n+0L4bswu8WstjbXCuowSZ/Mx7jbIv519EkZ+lfOupadDqn7WCW87OqpcQ3AKHndFarIo+hKDPtmgD6KHIzS0i5A560tABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRSFgOtAATg9DWV4h8RaZ4Y0WbVtVuPItIsAsFLFiTgBQOST/8AX6CptY1jTtD06XUNVu4rW0hGXklOB9B3YnnAGSemK+eLW68RfHXxelndyT2fhi1laZlhjwsaj7oJ6NKQcc5xlmC4yCAF6vi344+IruTSnnsvDMTrCq3M7LCMZIJUZDynJOADtyoJxg1734X8K6T4T0eLTtItVhjQYeQgeZM3952x8xPr0HQADAqbw/4d03wzosWk6VB5NpFnALFmYk5ZmJ6kkn+XQCtUDAxQAAEDk5NLRRQAxwSe2MYr5X1tJ/G/7Qslp5D7P7UW2eLzQcQwHEjDdxysbvj1OOe/1Q55x19q+bfh5pEGl/tB39lqc9+LmCW8aya5yZLp/mAZzt5zEZH3cAnBHXBAPpLOB7/lXyr4XvB4k/aHiv7SUTwz6rNNDIQV3RKHZTg8/cX0r3j4m+MU8F+Drq9jmVNRnBhsVJ5Mp/iGQR8oy3IwcBcjcK84+EWg23gbwbf/ABF1yV40mtSsMKqQyxbx2bALSOq7e2MHPzcAHutxPDbW0lxcTxxQRIZJJZGCqigcsSeAMc5r5y0Frj4yfGB7/UEd9B0smWOMwjyxCr5jjcHIy/VgSSQGAwAAMHxx8Zta8ZaZc6Qtha2Gl3DoxjXMkrBcHaXOARuAbhQeAM4zn6G+Hvhiw8L+DrC109cGeNbm4csT5kzIu5sEnaPlHA44/MA6mONYo1jRVVFACqowAPQCn0gGBS0AFFFFABRRSFscd6AMrxJr9j4X0G71nUWcWtqoZ/LTczEkAKB6kkDnjnkgZrx34S6Fc+LvG+qfEPV7N0geVzp6y8jeTjKnjcI1GzO3BJ7FeLPx41a61W50bwToryT317L5lxaxNjcMgRB+2CdzcnA2gnsa9U8M6Db+F/DNhosDrIlpCELhAokbqzbRnG5iTjJ696ANlPu06vK9R+PnhfStSubC50zXVmt5CjbrVI8477XdWAPBAIB5HFelaZfpqmlWmoRxSxJdQpMscy7XQMoYBhzgjPPPWgC1SHv9KWkPf6UAeQ/s4/8AJPdQ/wCwrJ/6Kir1+vIP2cf+Se6h/wBhWT/0VFXr9ABRRRQA1u/pjpXzz4yu5fCn7SGnaqDtS9+zeaZDlTG48hjgegUkZz8wB9BX0ORzXhv7Q+ix20Gi+K7eTy76CcWh+XO8YaRDycDaVbtzv5PAoA9yXpS1Q0TVbbXNEstVsyxt7uFZo92NwDDOGwSNw6EdiDV+gAooooAKawyadSGgDwn4IG00P4h+NvDEPnuY5T5Mj4OY4JnjO48fMfMXoMcHpxXu9eAfC4EftCeNcg8i+7et0le/A5oAWiiigANfP/8Azd3/AJ/58a+gDXz/AP8AN3f+f+fGgD6AooooAKKKKACiiigAooooAKKKKACiiigAoopCQDigALYOMGsfxL4k0zwvo02q6pcLFBEDtXIDyvgkIgJG5jjpn64ANReKvFujeENJbUdXuRHHgiONRukmbGdqL3J98AdyBzXg1lD4t+OXiJLi/wDPtPCkN2Q6QOpS32qWCgEgvIQQNxBxvJAA+WgAmg134/eKWmgEel6Hpsflo8h8zZu68DG6RsZOMABQM9N30PpGj2eh6Pa6Xp8SxWltGI40HHHcnHUk5JPcknvUeh+H9L8Oaetho9lFaWqkkog5ZuBuZjyxwBySTwPStQUAIAR/SloooAKKKKAEZd3HauC8c/CvTvGN9b6tDfXGl63bhRFewc/dYEFlBBJHOCrKRkZzgAd9RQB5PpPwStxrkeq+K/Ed74klgCiGO6BCbRk7X3M5ZckHbkDrkEEitb40S2qfCrWIrq5jgM3lLFu6yOJFcIB3JC9uwJ6A49Ab7w6V83+K9Tvfix8VoPDFhdPN4dtZ1Ja1ICbAB5sxblSeSqk5HK4A3HIB4tg8cf8A18V9xeEzbDwdoYtM/ZvsFv5Oc/c8tdvXnpjrzWdr/gPQ9b8Hv4Z+xQ21mik2giQgW8uGxIACMkFiTzzk5zk15l8DvFM2l6lf+AdXdVntpZBZ4HG5SfNj3Z9QWAx/fyegoA93opqHI/GnUAFFFFABVHWL9NJ0e91KUZjtLd52HqqKWI/T/wDXV6ue8ef8k+8S/wDYKuv/AEU1AHjXwC02bWvFeu+LtRTzp1JVJmhAUzSsWkZTjCsBwQuOJPQ19BDOBk8/TFeR/s5jPw/vzn/mKyf+ioq9C8ZanJongzWtSgmMNxb2UrwSbd22TadhwQQfm29RigD51Y23j79oseULY2cmojgkTRTxW6c9sESLEeOnz45619Tr06187/s36Ssup65rL+YHihjtY+RsYOSz5GM5GxMfU9e30QoIGDQAtIe/0paxPFPirSfB+jSapq87RwL8iKi7nkcgkIo9Tg9cD1IHNAHnX7ORA+HuoZ/6Csn/AKKirrPG/wATtD8BXFnBqcN9PLdKzqtpGrbQpAy25l6nPTPQ5xxnyr9nzxbpOny3Xhq6gEOoX0/nQXWB++woHlE4yMYYjPB3N0ON1r9pO9GPD1iuxnzPMwMY3AfIFw5GQD83AIzgZHAwAeieEfi14d8Z3F9b2EOpQzWlublo5rbczxg4YoIyxJBKjHU7hgHnHOf8NG+EMZ/s7XP+/EP/AMdrr/hroFv4f+H2i2sEflyS26XNwWC7mmkUM2SAM4ztHfaqjJxXgngW2t/HXx0l1E2Rm003VxqLxybU2ICTGWAOCd7R5Azk5zkZoA+predLm3jnjEgSRA6iSNkYAjPKsAVPsQCO9c38RtHGu/D3XLA7yzWrSxhGAzJH+8QHPGCyKD7Ht1rp1xjjpTJo0ljaORVZGUhgwyCDxgjv9KAPK/gB4ittS8Af2QqlLnSZWWQZJ3JIzSK/QAZJdcZJ+TPcV6wDmvnj4CpNoXj7xP4cuXBmhiYSGPlC0MmwkE4PVzjjoTX0MpyKAFooooAKQ57elLSZoA+cdGwP2q5VwP8Aj7uf/Sd6+jlGM/qfWvnHRv8Ak62Q/wDT3c/+k719HDmgBaKKKAEJwcc14Bc/6F+1nDcXf7iGfb5Ly/KJN1p5Y2k9cuCox34r38jPevAfiiMftB+DP+3H/wBKnoA9+U5GaWkU5FLQAUUUUAFFFFABRRRQAUUUUAFFFJkZx3oAM8471i+K/E2neEvD9xq+pTKkUQxHGThppMHbGo7k4/AAk8A1T8Z+ONF8D6bDfas8rGV9kNvAA0sh7lVJHAHUkjGR3IB8V03wx4h+NfiWPxRqzxWfh4TmNYRMWZI1x+7Qep6FjjkkgcBaAI9C0rWvjv4ln1TXrmaz0SwAjiS3Q7FJwTHGWyNxGCznJ+7xgrt+iNN0yz0rTbfT9PhWC0tkEcUS9FUDjryfrnJPJqSzsbexto7a0git7eMbUhhQIijPZRwPwqwBgUAKKKKKACiiigAooooAKQkDrS1T1S/g0rS7vUrksLezgeeUquTsVSxwO5wDQBwHxn8cXHhDwrHDYYW+1Pfbo5Yq0SbfndcEHcNy4PYnPtUXwU8EW3h3wfb6xNAh1XVIhK83UpCeUQHsCMMfUnBztGPOfAVjcfFz4o33iLXVE2m2JEn2WZvMVVJbyYQBjKjBYnaQxQ7h85r6UXpQAbTz0rw/4x+HZfDGuWXxJ0Wdkv4bmJLiEk7ZMKQGLbgcFVWMoPvA/Un3Ks3XtHs/EGi3mkagm61u4mifABZc9GXIIDA8g44IBoApeDfFNr4x8L2et20TQi4Vg8LMGMbqSrLkdeRkHAyCDgZxW+Dmvnz4daxefDT4hXfgDWrgvYXM4FnMIesz7RG3ByFdcAj5trY5HzGvoJfuigBaKKKACsPxnKIPBGvzNGsgj024co/RsRscGtysHxqjP4I15VjMrHTbnEXJ3/um4wvPPTigDzn9nK4T/hBdShIcOupM5JjYLhoo8YbGCflOQDkcE4yMz/HjxhY6d4Tm8NpKx1PUBE3lgspSHeSX3Y2kExlduc/Nnp1zP2b7jf4f1u2N7v8ALuo5PsnlY8rcuPM3/wAW/bjb28vP8VezXGn2d7t+1WkE4XO3zYw2PpkUAeO/s3z2T+F9ZgjbN8l6rzLg8RMgEfPTqsnTnj6V7WPf1qvb2FtZqy21vDAG+8IkCbvrgVZAxQAVwfxb8H33jTwZ9i02SMXttcLdxRycCYqrLs3fwkhzgnjIAOM5HeU1hkHpQB8n/BPwvfa747h1GDy0stJcTXEjHk7gwVAvctg+wCnnOAdD48ajJrHxKg0izhnlls7eK3ESxgmSWQ7/AJMZLZDoPqDj1Ov8BdY0vw94e8XavqFy8cVt9mMwKcBf3gTac5ZixI24GMLyc8cb4T8T6VN8Yh4p8QOltYvdT3bCVWn8ssrmMDapJKsVwccbc8cUAei+NvA+veCvAl1rFt8QfEM0lksS+V9pkRGy6x8AP8oG7IHPTFN/Zw0BTb6x4jkSMszixgIZtygYeQFemDmLB5Pynp3n+PvjLSn0RvCtvc+Zqi3cb3UXlsBHHsLj5iMEncnTPfpiui+AL2z/AAyQQSbpVvJhcDZt2vwcZ/i+Uoc++O1AHqAGBSMCcU6igDwDxqv/AAh37Q+h6/JxbagYvMmm+SGPKm3k+bp8ibXPpuBOAa9+XGMjHPpXif7SNgJPDuiajsyYbt4Mg4Pzruxjv/q69J+H+qf2z8P9BvjO88j2UaSyyPuZ5FG1yTk5O5W5oA6SiiigApDS0hPIoA8C+HRTUP2i/F01xBE724u/KJUEoVnSMEeh2kjPoSO5r30DGa+ffhNKtx8e/GFxGHCSpeuodCjYN1GeVYAqfYgEV9BCgBaKKKACvnn9oOK40fxj4b8S2l1suRCUiHlg+U8MnmK/OQcmToR/D3zgfQ1eG/tJxqdH0CbjctxKgGBnBVe+M9u3Hr0FAHt8TpJGskbh0YblZTkEHuDT65r4ezxXHw68OPE+9RpsEZI/vKgVh+BBH4V0tABRRRQAUUUUAFFFFABRRSFgCB60AGRnFct448eaT4G0Zr6+fzblwVtrNGw87/8AsqjuxHHuSAc34lfEnT/AmmBQFudYuIybW07Ac/vJO4QEexYggdCV8g+Hnw6v/iZqkvi3xVdXEljJcZfcCHvSOCFPAVBwuV6BSo24yAB/hPwb4g+LPij/AISPxhHeDSZIzsnUiINtxtjjUgkJ8x5AwcN8xbJr6K0rSrHRNMt9N0y2jtrO3XZHEg4A6n6kk5JPJJJNWYoUiRUjVUVRtCquAAOgH0qUcUAIOlLRRQAUUUUAFFFFABRRSE4oACQK+evjf4sv9a8TW3gXSpUa3d4RMIiS0lwzHCNtJyoyh24zu57CvWfiL4x/4QnwlPq8UCT3RdYLWORsKZG6Z7kAAkgdduMjqPNfgH4QkuGu/HGqlp7m4d47VpsOWycyTbjk7icrnIP385zQB6v4O8K2fg/wza6PZbXEQ3TTbNpnkP3nYep9CTgADPFb4GBihQQOaWgApCue9LRQB5N8avAS63ox8S6eZU1nSotwEbYEkKsWIGSNrLlnBHJ5GDkY3/hX47Pjjwp9pu2iGqWshhu0jAUZ6owXcSAV7nGWV8DArt5YlmRkkVWjddrqwyGHofb/ABr5ytYZfgV8Uo47iVrjw7qke37S8TFki3dyBy8ZxnGcq2cAsAAD6QBzRUVrPFc20c9vLHLDKoeOSNgyspGQQRwQRjBqWgArM8RRRzeG9UjlRXja0lDKwBBBQgjmtOs/Xv8AkX9S/wCvWX/0A0AeH/s0Hd/wk/QY+yf+1q9/AwK8A/Zm/wCZo/7dP/a1fQFABRRRQAUh7/SlpD3+lAHz5+zfbw3dv4ttriKOWCVbZJI5FDK6nzgQQeCCCRg17VY+EvDumXaXdhoGl2lzHnZNBZxo65BBwQM9CRXjP7M3XxR/26f+1q+gKAMrUfDOh6xcLcano2nXs6oEWW5tY5GCgkgAsDxkk49zVrTtMsdJtBa6dZW1nbglhFbRLGgJ6nCgCrdFABRRRQBy3xG0f+3vh7rmnhJ5JHtWkijgGXeSP94igYOcsoGAMnPFcb+z94hTUfAkukOyCfS7hlCKjA+VIS6sxPBJfzRx0CjjufWG6n6V87fDV08JfHrWPDlurrY3bzwQwxykogUmWMtnkkIrLk8/MecE5APoyikHSloAKq6hfW+mWFzf3koitbaJpppCCdqqCScDk8DoOatVx3xT1T+yfhl4guPJ87faG327iv8ArSIs59t+cd8YyM5oA8n/AGbLGCTVfEGotuM8EMMCHPG2RmZv1jWvogdK8g/Z208weA728e18qW51B9szR4MsaogGGx8yhvMHXAO7vmvXwMCgBaKKKACvKv2gdKe/+HIu4/KH9n3cc0hcfMUbMeFOP7zqe3A9QAfVayvE2nTax4X1fS7dkWa8spreNpCQoZ0KgnAPGTzgGgDlPgperd/CjRx9oWaSAywyAMCYyJW2qQOh2FDj0IPeu/ByK8K/Zv1pX0zWtDdoleGZLyMbvncOux8L6LsTkd3Gete6jpQAtFFFABRRRQAUUUhYDrmgAJArhPiX8SLHwHpQEey51m4X/RbQnoDn9446hAQfdiMDHJVPiT8S9P8AAWnoFVbrWJ0JtrTdjA/56SdwnB46sRgdCV8y8A/Cq98eyy+LvG8900d46ywx79j3S8fM3HyxkAKoXaccjACkgEHw5+Hmp+Ptfl8V+M0knsZVEsbO6g3UmQANoH+rCjGBt/hA4Br6LtreK2t44IIkhhiQJHGihQigAAADgADsKIbaK2t47e3ijigiQJHEihVRQMBQB0AHb2qUUAApaKKACiiigAooooAKKKKACmOBkcZNOJxXlfxt8e3HhbQ4tL0uSaHVNRU4njGDBEMbiG7Mc4GOnJyCBkA861K7uPjV8W4dIEpj0OxMqpLbDcTCp5l5xzIQgBxhdy8HBz9H6bp1npOnxWGn20VtawjbHFEuFUfT65J981w/wk8CQeD/AAqlwZGk1DVI4ri5YjGz5crGBkg7SzfN3zn0A9CUYGKAFooooAKKKKACuA+Lvgs+MvB8gtYg+q2JNxZnOC39+POCfmHQcZYJk4zXf01gT6UAeXfBTxemo+GV8M6i0kOt6Pvhkt5gwfyVIAJBUbdufL25JGwZ616kDmvnDx7ZN8MvjTp3ixFd9PvpmuW8tEDAnKzoBnk7X3AnGd+M5Bavou2uIbq2jubeWOWCVBJHJGwZXUjIII4II7igCWs/Xv8AkX9S/wCvWX/0A1oZzWfr3/Iv6l/16y/+gGgDw/8AZm/5mj/t0/8Aa1fQFfP/AOzN/wAzR/26f+1q+gKACiiigApD3+lLSHv9KAPAP2Zuvij/ALdP/a1fQFfP/wCzN18Uf9un/tavoCgAooooAKKKKAEIJNfOPj7Tx4O+Pmj647mOwv7iC5eaRGEcfzBJQGzyQBvPpvAxjr9H145+0RoL3/hKx1mISF9MuCsmHVVWKXCkkHknesQGPU5HcAHsKYCgDtTq5P4Z61Nr/wAOND1C53+e1v5UjSSF2doyYy5J7ts3fj1PWusBzQAV5L+0Lqctn8P4LOGcIb29jSVMAl41DP8Ahh1jPGD+Ga9ZzivB/wBoS5fVdV8MeGbALNfzu0nkH5SS7LHF8xIXBIcdeNvOB1APR/hRai0+F2gRLEYw1t5u05/jYvnn13Z/Guyqhommpo2hWGlxyNKllbR26yMMFgiBQT7nFX6ACiiigAprDmnUhGaAPnHwzLJ4R/aU1LT9kaw6hPNCQExsSUecm0A4HIQc5GM8en0cvIrwP41WTeF/H3hrxra27OgkQXCxR+WHeJwwDyDPLoSvIziPuOnu9rNDc20dxbzJPBKokjlRwyup5BBHBB60ATUUUUAFFFIWA60ABIBA9a4T4l/Eqw8CaUUUx3Gs3CE2toSenTe+OQoP0LEEDuVz/i38TW8C2cFlp0Ky6xeKWjMgykEYPLkdyTkKOnBJ6YblfhN8LtQk1eHxx4nkhlluF+12sD/O5lc7hLJ2Bx8wAycsD8pXFAEXw3+G154uvIfHfjSe4uZZ5TNDaTpjzgMbJG9EznEYABAX+Hg+9AYGKFG0YpaACiiigAooooAKKKKACiiigAoopCwBxQBna9q9poGiXmrX2821pEZXCLljjsASOT0HT614L8PdBl+K3jzU/GHiK2EulwS4gtpGBUvwY4iNoDIiHJ6biVJDZYVc+MHii/8AFniy2+HOhh1H2iNLqTedksjAMAQuT5aA7mJzyOg2ZPsvhXwvpnhDQYdJ0lGFvGSzSPgySOTyzkAZboM44AA6CgDaXpS0gGBzS0AFFFFABRRRQAUUUUAc3478Nr4t8H6jo48oTzRZt3lB2pIDlTkcjkYJHYng9K8y+BPiI6bNqfgXVpGi1G1uZXhhlcMBtOJI0O4g4YM2B6seeTXuBBPIrwL44aHe+GvEem+PNAM1tctIEubiJs7JFULGSCTwyAqRjacYOd3IB76vTrz3qhr3/Iv6l/16y/8AoBqHw3rUOv8AhrT9XhdGS7gSU7DkKxHzLwTgqcgjsQe9S66c+HtSI/59Zf8A0A0AeJfszqBH4mbepJNsNvOQB5vJ7YOf0Ne/Cvn/APZm5Higf9en/tavfx05oAWiiigApD3+lLSHv9KAPAP2Zuvij/t0/wDa1fQFfP8A+zN18Uf9un/tavoCgAooooAKKKKACud8daPNr/gjWdLtV33U9qwhTj5nHzKMkgDJUDJOBnNdFTW+8D6UAeL/ALOV6jeE9WsABvhvvOb5ucOij7vb/VnH/wBavaR0rwT4bovgz45eI/CYnjjs7tWNtBHlwSuJYl3EZBWJ5AcnGQRljjPvSHK5oAG+8B3rwGIf8Jv+0y7Nmex0MkhH/dmPyeBjGC2Lh8jOcjrxxXumqXiabpt3fyEeXbQPK25tq4UEnJ7dOvavCv2b9Fl83WtdeNRDhbOJ+CS3DyAc5GB5fbBz14oA9/X7vf8AGnU1TleetOoAKKKKACiikJxQBwXxl0aLWfhpqe+Nmls1F5EVDHaUPzEgHkbC456Zz2rN+BXiM6v8PIrKdpGuNLla2LvIXLRn5kOT0ADbAMnAQfQej3ltbX9pPZ3Uay288TRSo3RkYYIP1BrwD4eTT/Db4u6l4T1Frq303UXKWQm2kStv/cSbhj7yhl+X+IgEcfKAfQ4IPTt1paRcAcdKCcUABOOteYfFL4s2fg+3l0nTJPO16WIldoDLa5HDPnjdg5VcHsTwRmh8VPjDH4cX+yvDVzbXGr73juX2lxagAj1x5m7HB3AbSGHIFVfhb8LrlrseM/GSyz6vPJ58Ftcg7omzkSyA/wAfov8ACOeuAoBB8L/hCxLeJfGtqLu+usyRWd4N5XdndJMrA7nOfunOMkn5sbfb1GBQBiloAKKKKACiiigAooooAKKKKACiiigBCcVxnxK8dJ4D8MnUEgFxezv5FrCxwu/BJduclVAycd8DIzkdhM6RI0kjqiKCzMxwAB1J9q+cEa4+NvxYeN7gf8I1pbmSONwQGgBUELgK26UrnkgqCf7oBAOo+CHgKWKP/hOdYuPtOoX6ObZJPnZAzfNKzEE+Y2D0P3WOSSxC+1KMD/69Q2VpDYWUNnbRLFbwIsUUajAVFGAB9ABU9ABRRRQAUUUUAFFFFABRRRQAVQ1fSrHW9On03UrZLmzuE2SwuOGHXtyCDggjkEAgg1fpCM0AfO3wq1C9+H3xK1HwLq8qiC6kCxuz/Is2MxsPmwPMQgY+8T5Y4xivedc/5F7Uv+vWX/0A15h8dvCc97o9r4q0pJxqukuC0sBIdYASdwwN2UbDZBGAWP01vA3jY+NfhbeT3dxFJq1tbSw3qooU52ttfb/tLg5GASGwBjAAOJ/Zm/5mj/t0/wDa1fQFfP8A+zMMHxR/26f+1q+gKACiiigApCece1LTWz1FAHg37N1vNaXPi23uIpIZ4ntUkjkUqyMDMCCDyCDxive68E/ZunlurnxZcTyvLNK1q8kkjFmdiZiSSepNe90AFFFFABRRRQAUhGaWigD56+OUr6D8TfC3iMwtOsEcTiI/IHMM28jfzyd4HTjg85r6BhkjliWSJxJG4DKynIYHuCOoryT9ojRPt3gi01WO33zaddDfLvx5cMg2txnBy4iHQn04zXcfD/U4tX+H2gXkUxnJsY45HIIJkQbH68/fVhnocZ6UAcl8ePFCaN4EOmQuwvNWfyU2MVKxKQ0pyBggjahUkZEh6gEV0/w48LN4S8C6dpc8caXmwzXeEUEyudxDEE7ioITdnkIMYHA8Z8WxS+PP2hrbT7W1N/Z6fNBBcBeVWGNw0xY8YAZnX64A5IFfSCkKMcY68elADhS0m4ZxS0AFFFFACE4OOa8u8f8AxntvBOttpC6DeXd0gV2eWQQROjDIZGwxbnKngDINeon6nPtXzH+0H4hGo+MLXRogpj0qDLkqQ3mS7WIyTgrtEfQDkn8ADp7T9pKxe9jF34buIbUj55YrpZJFOM4CFVB+bj7w457Yrm/jTc6f4s0rQfG2jSebZTK1hcFiA8Egy6o65+9gv04wByQymuz+I8ul+EvgjB4Su7mA6obW2hW3t5vmZ1ZWaTB+by90bHOB6cZ4ofBHw2Na+GuvWWq20g03VJjHGzKPmwoBdNwI3KwBDYOGX1WgD1/wxrUfiHwvpmroYv8ATLZJXWF9yo5HzJn1VsqfcGvN/ix8WZvDl43hrQbfztVlixNOSf8ARy4+UIo5MmCG9BlfvZOPFbHxz4s8F6Xqvha0vDbxi4dHcLiSBxlHEbHlc49MjGRtOc9H8LfEvgTwuJtX8RG5udekkPlubfzFgX+8v+2TnLHkdBj5sgHc/Cj4MyaJdJr/AIohiOoROTa2WVcW7A8SMwJUt1IAyBwfvY2+2AYFeYD4++Bv+e99/wCAp/xpf+F++Bv+e99/4Cn/ABoA9PorzD/hfvgb/nvfn/t1P+NH/C/fA3/Pe+/8BT/jQB6fRXl//C/fA2f9fff+Ap/xo/4X94G/5733/gKf8aAPUKK8v/4X94G/5733/gKf8aP+F/eBv+e99/4Cn/GgD1CivL/+F/eBv+e99/4Cn/Gj/hf3gb/nvff+Ap/xoA9Qory//hf3gb/nvff+Ap/xo/4X94G/5733/gKf8aAPUKQsAcGvMP8Ahf3gb/nvff8AgKf8agvPj/4NSznktTeT3Cxs0ULQlBIwBIUtzgE8ZwcZ70AZ3x38cvY6bH4S0o+Ze6in+lGNm3xR7htUAcEucgjJ+UEEfMDXafDjwDb+A/Dv2RXE1/cMst5NgYL7QNqnAJQHON3dmPGcV88+EPFelT/EefxZ41lkkKs80MKxmUCVjhQA2cIiklRnIKpjpXtK/HzwMo/1991/59j/AI0AeoAYGKWvL/8Ahf3gb/nvff8AgKf8aP8Ahf3gb/nvff8AgKf8aAPUKK8v/wCF/eBv+e99/wCAp/xo/wCF/eBv+e99/wCAp/xoA9Qory//AIX94G/5733/AICn/Gj/AIX94G/5733/AICn/GgD1CivL/8Ahf3gb/nvff8AgKf8aR/j/wCB1jZlk1ByASEW25bHYZIGfqaAPUaK8qt/2g/BU0ZaVNUt2BxsltgSff5WIqX/AIX94G/5733/AICn/GgD1CivL/8Ahf3gb/nvff8AgKf8aguf2hfBcG3yotVuc5z5Vuo2/Xc46+1AHqNzBFdQyW9xFHLBMhjkjkUMrqRggg8EEZGD618p+IV1L4O+NtV0vR7yZ7G9sioWbB8xHRgrMMYLI+4ggdsdCRXqs37RPg6OUqtnrMowDvSCPB/OQH2/CvP/AIo/EbwX4+0VPsum6vFrNqR9lnkjjVdpI3q+HJK4yRxkEDkAtkA3v2Zzk+J/+3T/ANrV7/Xyj8IPiNpHgA6ydVtr6b7cIPL+yojY2eZu3bmX++MfjXqUf7RPg5xIWs9ZTapYboIvm9hiTr9cUAeuUV5Ov7Q3g1rZ5Db6usitgQm3Te3uMPt/XtUP/DRng/BJ0/WxgZwYIuf/ACJQB69SHv8ASvIf+GjvB/8A0Ddc/wC/EX/x2j/ho3wgc/8AEu1z/vxD/wDHaAOe/Zm6+KP+3T/2tX0BXyj8IPiNpHw//tk6rbX0wvvJ8r7KiNjZ5mc7mX++Ome9en/8NHeEB/zDdc/78Rf/AB2gD1+ivIP+GjvB/wD0Dtc/78Rf/HaP+GjvB/8A0Dtc/wC/EX/x2gD1+ivIB+0b4QP/ADDtc/78Q/8Ax2l/4aM8IjOdN10Y/wCmEX/x2gD16ivIP+GjfCH/AEDdc/78Q/8Ax2lH7RnhEjP9m67x/wBMIv8A47QB6Zr2mf21oGpaX5vk/bbSW383bu2b1K7sZGcZzjP5V8x/Df4pzeB/DOt6WtjbzuwN1aO77f3zbIyrD+JdoDcYPykfxZX05v2jPCGf+QdrmP8ArhD/APHa+fY7CbxR4ua10W0kLX925t4jyVRmJ+bA4CjkkcAAntQB7t+z5osjaXrHiq+fzrvULjylklUNJtX5nbzCSTvZhkccxgknt2Hjn4p6D4GHkTlr7Um3bbO2dS0bbQy+bk5QHcMcE4PAOKyPFWp2Xwl+E1vpFlcZvWtza2ZRtjvMwy84BLEAMxfAyASq8ZBrlv2evDUzvqfiy/gkZpv3FpNLnL5JMrDI55CjcO4cetAGhF+0FFZarLZeIvCWo6Y0afMiyeZKGOCA0brHgFTnP04r1/SNSh1nRrHU7dXWC8t47iNZAAwV1DAHBIzz6muK+LPgS98d+HbO0037El9b3Yk865yMRlWVlBCsRk7DjgHbz0Fd7bQQ2ttHb28KQwxKEjijUKqKBgAAcAAdqAJaKKKAGv0zxgdc9K+QtJhm+LfxazfyS28eoTPLKYlDNDCiEquQAOFVU3EdSCQelfTPjzV7jR/C9w9ppGo6pcXINukNhCZHUsrfO2OQox19SPWvmT4ceL4Phx4nv7vU9Mu5ZzbtaGAYjaM71Lbg3OQUA/OgDW+KXgDTvhrqmhXWlTy3kNwzu0N+iSDdEyH5sABlYOBtI7Hk54+hvAOtReJPAmj6pDbJbLLAEMEcYRI2QlGCLk4Tcp289MV4d4lTxZ8bPEOltZaBeabo8ELGKe7XEWGYb5N+0biRsGwFvu8dTXvvhTw/beFfC+n6JandHaR7GfBHmOTl3wScbmLNjJAzgcYoA8T/AGg/BcVrJb+LrNSJLiQW98C5OTtxG4GOPlUqef7mBksTp+BvhV4E8X+DdO1r7NeLLPHtnjW5ICSqdrgDJwMgkZOcEV6z4q0C38U+GdQ0S5bZHdxbRJgny3B3I+ARnawDYyM4x3rw34Y+Jbj4aeL77wJ4k8tIri6TZOkgMcUrINrZC5KyL5XJI24GQPmwAd2PgH4GP/Lvff8AgS1Tx/AnwEls8R0yeR2ORK13JuX2GCB+lekLjBx60tAHl4+AXgfjMF6cHP8Ax9HmtAfBL4fAf8gDJxjP2yfn8nr0CigDz1Pgh8PlLltCZ9zZAN5ONox0GHHH1yead/wpL4ef9C//AOTtx/8AHK9AooA8/wD+FJfDz/oX/wDyduP/AI5R/wAKS+Hn/Qv/APk7cf8AxyvQKKAPP/8AhSXw8/6F/wD8nbj/AOOUf8KS+Hn/AEL/AP5O3H/xyvQKKAPP/wDhSXw8/wChf/8AJ24/+OUf8KS+Hn/Qv/8Ak7cf/HK9AooA8v134WfDPw/ot1ql34cnkht03FILm5kkc5AAVRJ1JIHOAM5JAya8x/4SH4HD/mTdc/7+t/8AJFfRuv6l/Yvh7UtV8ozfYrWW58oNt37ELbc4OM464NfPX7OWlNP4q1bVSIjFZ2aw4YZYPIwII4x92NwTkfe75OADd8Jab8EvF9z9ks9ImtL0khLa9uZo3kAx90iUqSc8LndwTjAzXcj4J/Dwj/kX/wDyduP/AI5Xh3xtdT8WrpbHP2lI4FPlAh/M2qVxjknBXH4enH1enTrmgDgf+FJfDz/oX/8AyduP/jlH/Ckvh5/0L/8A5O3H/wAcr0CigDz/AP4Ul8PP+hf/APJ24/8AjlH/AApL4ef9C/8A+Ttx/wDHK9AooA8//wCFJfDz/oX/APyduP8A45R/wpL4ef8AQv8A/k7cf/HK9AooA8//AOFJfDz/AKF//wAnbj/45S/8KT+Ho6eH/wDyduP/AI5Xf0UAef8A/Ckvh7/0L/8A5OXH/wAco/4Ul8PP+hf/APJ24/8AjlegUUAef/8ACkvh5/0L/wD5O3H/AMcoPwU+Ho/5l/8A8nLg/wDtSvQK53xr4rtPBfhm51q7XzfLwkVuHCtNITwoJ/EnqQATg4wQDgfFHgj4P+D7JrjWNMSN9haO2S9naaXHZU8znnjPAHcjrXHaRf8AwN1bVbawPhrUrPz3EYnuriRYkY9NxWckAnAzjAzk4GSIfg7plz45+JV94s1i4jnexYXDpKN5aWTcI9oOcKgUkc/KVTHt2f7RkETeBtNuDChnTUljWQjLKrRyEqD1wSqkj/ZHpQB5z8W/AFh4D17Tb7S7bzdIu8/6NPI7BZEOWQkYYKVK4+Ytw/IwK9a0T4WfDTXdEsdVstBJgu4FmQG9nyARnBxKRkdCATggisbwXax/FD4Ftot+224snNpDcMq/I8YDRMMDgBWVCepAbnLZrM+CHia80TW7z4fa5G8NykrtbK+WKOoLSRZBIC4BcYwM7zk7hQB3n/Ck/h6Rx4fH/gZcf/HKP+FJ/D3/AKF//wAnbj/45XfL93rmnUAef/8ACkvh7/0L/wD5O3H/AMco/wCFJ/D0f8y//wCTlx/8cr0CmnrQB8x/A/wV4e8Zf28Nd0/7WbX7P5WZnTbv8zd9xgT90dc169/wpP4enr4f/wDJ24/+OVxvwA05tH13xxpjyLI9ncQW7OvRijTrkflXuVAHn/8AwpL4ef8AQv8A/k7cf/HKP+FJfDz/AKF//wAnbj/45XoFFAHAD4KfD0dPD/8A5O3H/wAcqzf/AAj8CajcGefw7bI5AGLeSSBeP9lGAz745rtqKAOfHgTwj38LaH/4L4v/AImqFv8AC/wRa6lLfx+GrAzSjDLInmRjkH5Y2yinjqAD19a6+mt1wOuKAPOvGXhb4f8AhvQdT8QX/hrTf3dm1usSR+UJS2dqqFGFctx5gXco74Feefs++DbiXVJvFd3A8dtCjQ2RdOJXOVd1J5woyuccliM/KRVT4palP8SviZY+EdDUO9hJLbCSRtimTgynkZwvl++dvAORn6G0jTY9K0Wy02EgxWlvHbocdVRQo9ew9aAPm/8AaKJ/4WBYcEf8SqP/ANGy17j8M9Ng0r4aeHbeBnZXsknJcgndKPNYZHYFzj2A61yvxj+GM/jGKHWdLeManY27o8Tbs3MY+ZVUg4Vgd+OOS/JGK5zwv8e9D0jwhp+m32kaj9rsbVLbEOx0fYgUNuLArnGcbTj3oA9um1TT7e9gsJ762jvLgEwW7yqskmBztUnJx7Crg7/WvnHwFZ6t42+Nb+KdW02aytxF/aCrJEwQxsvlwBWwAfUHGG8t/evo1Pu9c0AOooooAQjJFIFxnHGeeKdRQAzZyTgDPpTlGBg0tFACEZ//AF15B8bfADatpY8T6RDJ/bFlgzeU7AyQDPIX++hwQcjjd1IUD2Co5YknjaORVZGBVlYZDA9QR6UAcP8AC74g2/jbw0hnfbq9mix3kbFcyEAfvQFx8remBg5HQAnuwQen0r508R6ddfAvxxb65oZa50TUxIj2TucqBgmMuQehIZW68MDkZLe+aLq1trei2WqWm77PdwrNGGxuUMAcHBIyOhAJ5FAGhRSA5GaWgAooooAKKKKACiiigAoopCQOtAHl/wAfdSNj8MpbYQmT+0LqG2zuxswTLnpz/qsY465zxg8N8Ovhb41fQxfWviaPQbHU4BODbDzLhwfubtu3AKksCHJXd0BJxmfH7xVba34otdHghnQ6P5iStNEY97yBCdueSuFGDgZzkZGDXvfgPULDU/AeiXOmFvsgtI4kDAgqUGxlOeeGUjPOcZyetAHJ+Fvgnoei3jalrNxLr2qtN532m53KobJO7ZuO4knJLFskDpXpqjAxS0UAFFFFABRRRQAUUUUAFFFFABRRRQAV4T+0re3Edh4eslk/0WaWeaRMDl0CBTnrwJH9ufpXu1eWfHDwTfeK/Dlne6XFJcX2lu7/AGdGwZYmA37Rj5nGxSB6bsAnAoA1vhBoC6B8NNLUhPPvV+3TMjMQxkwUJz0Ij8sEAAZU/U+c/tJ6orXGg6RHcyb40lup7cFgpDFVjc9ifllA6kc9M89f4I+Kng23+H+mx3msR2k+mWUFvcQTRt5gcL5fyKAS4JXPy5wCCwXpXIaD4cuvjB4+m8Z3UX2HQbWdUt1Zf3l0I2BCn5uOPvMDjPAyckAHqHwt8OXHhj4eaZYXkEUN8wea4Ea4JZ2JG7gZYLtU+m3A4Arzb4z+DNQ0TXIPiB4dIilidGuxEnzJICcTYOVKkAKwwBwM7tzY96Xp0xnmqOt6Vb65o97pV0G+z3kDQuUxuUMCMjIIyOoOODigCl4Q8U2Pi/w1aaxYuhEqjzolfcYJMDdGcgHKnjOBkYI4IrcByOK+e/CF/cfB34kXPhTWpbcaLqREsN6742D5ljkOBxkrsYEcHnOAS30IuMZHegBaawz/AEp1IaAPGvgrAbXxt8RrYzSzmHUUjMszAvJiS4G5iAMk9ScV7NXj/wAIf+Sh/E3/ALCo/wDR1xXsFABRRRQAUUUUAIWA61wPxc8YReFfBV2sV6bfVb2Mw2QQkOTlQ7AgHBVWznjtg5xXY6tqljoumXGo6lcx21nbpvllk6AfTuScAAckkAcmvnvwhp178YPiZceKtYtGj0W0KlYZcywkpjZbgsRkYy7YBGc5A3igDt/gd4CuPDmi3GsaraC31O+yiRurLLBCCPlYHoWIBxjoFzzwPWlXbmkQbRinUAIRmmhMY4Ap9FADApyCTzTgMClooAKKKKACiiigAooooAKKKKAMTxT4a0zxZo0ulatAZLeTlWXAeJx0dD2YZP15ByCQfE/BXiTVfhV44k8D+JJs6G8rfZrm4yixBiSsqdQI3I+YZwrEnIw+foYgmuG+JvgK18a+HJR5TtqtpHLJYGOTblyAdhyQuGKKCT06560AdyOBQDkV4x8IviPqd5fz+E/FryR6pbIv2eS7AjlkxgGJw2C0nII4JIDE9Mn2ccUALRQDmigAooooAKKKKACkIzS0UAN2n2pQMDmlooAKKKKACiiigAooooAKKKKACiiigAooooAKay7qdRQB81/FPw9eeHPi1beLrzTWm8Py3dtcO9ugcAJsEiODgBmKkjJw27rndj2vwv4+8M+MZZYdF1Jbi4jjEkkDxNG6g8dGA3YPBIyASPUZ6dlJqnp2j6do9u1vpdhaWUDOXaK2hWNSxABJCgZOAPyFAF0UEZ9KAMCloA8q+NvgK68WaBb6hpkfm6jpvmMYh1miK5ZVAUln3KuBkDlupIrc+FXjS38YeC7Zy4/tCxRbe8jLgtuAwJCBjAfGemM7gM7c125BJ68V87aip+D3xsGoCJv7C1ZTukfcfLikdTKNwXqjgNgZO0qCctmgD6KB3Dig02GRJYUkjdXjcBkZTkEHoR7U40AeQfCH/kofxN/7Co/9HXFewV4/8If+Sh/E3/sKj/0dcV7BQAUUUUAFISBQWA6+ma8V+LvxE1JdWg8GeEpPNv7lTHeNEgZ1LcLGhz8rY3FuMjK4IOcAHPeN9cvfjR4ptfDXhBlfTLKM3Mk1zmFWfO3zDnLFQGUABd2WbjFe76BoVl4c0G00fTgy2lrH5aZPzNzksSB94kknAHJNcz8M/hxaeAtHYF0uNWuQPtdwudox0RP9kc8nljyccKO6AxQAKMCloooAKKKKACiiigAooooAKKKKACiiigAooooAKQjJzS0UAeVfGD4YXHjWG01HSXt01OyiePypBt+0p1VA+cKQd2AeCXOSAKg+FfxZ/wCEkmg8N61A8OtRRFRMx+W5Kdcg4Ik25JGMfKx4yFr1ogk9a8i+Kvwtj1GA+IvCtqbfxFBMJ3Fq/lm45yWHIxID8wYYJ5zkkYAPXl6UteQ/Dz4prDF/wjPjm4lsfEcE4hVrqEr54blckDCsOmTgEFTkkmvX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proof	Geometry	Math	Chinese
71	"如下图, 设 $\triangle A B C$ 的外接圆为 $\Gamma$, 在点 $B 、 C$ 处分别作圆 $\Gamma$ 的切线, 两条切线交于点 D. 由 $\triangle A B C$ 的边 $A B 、 C A$ 分别向形外作正方形 $B A G H$ 、正方形 $A C E F$, 设 $E F$ 与 $H G$ 交于点 $X$. 证明: $X 、 A 、 D$ 三点共线.

<image_1>"	['如图, 设点 $A$ 在 $G F$ 上的投影为 $R$, $A R$ 与 $B C$ 交于点 $M$.\n\n<img_4112>\n\n由面积法知\n\n$$\n\\begin{aligned}\n& \\frac{B M}{C M}=\\frac{S_{\\triangle A B M}}{S_{\\triangle A C M}}=\\frac{\\frac{1}{2} A B \\cdot A M \\sin \\angle B A M}{\\frac{1}{2} A C \\cdot A M \\sin \\angle C A M} \\\\\n& =\\frac{A B \\cos \\angle R A G}{A C \\cos \\angle R A F}=\\frac{A G \\cos \\angle R A G}{A F \\cos \\angle R A F}=1 .\n\\end{aligned}\n$$\n\n故 $M$ 为边 $B C$ 的中点.\n\n设 $X A$ 与 $B C$ 交于点 $K$. 则由 $G 、 A 、 F 、 X$ 四点共圆得\n\n$$\n\\begin{aligned}\n& \\angle C A K=90^{\\circ}-\\angle F A X=90^{\\circ}-\\angle F G X \\\\\n& =\\angle A G F=90^{\\circ}-\\angle G A R=\\angle B A M,\n\\end{aligned}\n$$\n\n即 $A X$ 在 $A M$ 的共轭中线上.\n\n另一方面, 在边 $B C$ 上作一点 $M^{\\prime}$, 使得\n\n$\\angle C A D=\\angle B A M^{\\prime}$.\n\n由面积法知\n\n$$\n\\begin{aligned}\n& \\frac{B M^{\\prime}}{M^{\\prime} C}=\\frac{S_{\\triangle A B M^{\\prime}}}{S_{\\triangle A C M^{\\prime}}}=\\frac{\\frac{1}{2} A B \\cdot A M^{\\prime} \\sin \\angle B A M^{\\prime}}{\\frac{1}{2} A C \\cdot A M^{\\prime} \\sin \\angle C A M^{\\prime}} \\\\\n& =\\frac{A B \\sin \\angle C A D}{A C \\sin \\angle B A D}=\\frac{A B \\cdot \\frac{C D \\sin \\angle A C D}{A D}}{A C \\cdot \\frac{B D \\sin \\angle A B D}{A D}}\n\\end{aligned}\n$$\n\n$$\n=\\frac{A B \\sin \\angle A C D}{A C \\sin \\angle A B D}=\\frac{A B \\sin \\angle A B C}{A C \\sin \\angle A C B}=1 .\n$$\n\n故点 $M^{\\prime}$ 与 $M$ 重合.\n\n因此, $A D$ 也在 $A M$ 的共轭中线上.\n\n综上, $X 、 A 、 D$ 三点共线.']			['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proof	Geometry	Math	Chinese
72	"如图, 在锐角 $\triangle A B C$ 中, $D 、 E 、 F$ 分别为边 $B C 、 C A 、 A B$ 上的点, $L 、 M 、 N$ 分别为线段 $E F 、 F D 、 D E$ 的中点. 

<image_1>
证明: $A L 、 B M 、 C N$ 三线共点的充分必要条件是 $A D 、 B E 、 C F$ 三线共点;"	['设 $\\angle C A L=\\alpha_{1}, \\angle B A L=\\alpha_{2}$,\n\n$\\angle A B M=\\beta_{1}, \\angle C B M=\\beta_{2}$,\n\n$\\angle B C N=\\gamma_{1}, \\angle A C N=\\gamma_{2}$.\n\n由分角线定理知\n\n$1=\\frac{E L}{L F}=\\frac{A E \\sin \\alpha_{1}}{A F \\sin \\alpha_{2}}$,\n\n$1=\\frac{F M}{M D}=\\frac{B F \\sin \\beta_{1}}{B D \\sin \\beta_{2}}$,\n\n$1=\\frac{D N}{N E}=\\frac{C D \\sin \\gamma_{1}}{C E \\sin \\gamma_{2}}$.\n\n三式相乘得\n\n$\\frac{\\sin \\alpha_{1}}{\\sin \\alpha_{2}} \\cdot \\frac{\\sin \\beta_{1}}{\\sin \\beta_{2}} \\cdot \\frac{\\sin \\gamma_{1}}{\\sin \\gamma_{2}}=\\frac{A F}{A E} \\cdot \\frac{B D}{B F} \\cdot \\frac{C E}{C D}$\n\n$=\\frac{A F}{F B} \\cdot \\frac{B D}{D C} \\cdot \\frac{C E}{E A} \\quad \\quad (1)$.\n\n由塞瓦定理,知 $A D 、 B E 、 C F$ 三线共点的充分必要条件是\n\n$\\frac{A F}{F B} \\cdot \\frac{B D}{D C} \\cdot \\frac{C E}{E A}=1$.\n\n又由角元塞瓦定理,知 $A L 、 B M 、 C N$ 三线共点的充分必要条件是\n\n$\\frac{\\sin \\alpha_{1}}{\\sin \\alpha_{2}} \\cdot \\frac{\\sin \\beta_{1}}{\\sin \\beta_{2}} \\cdot \\frac{\\sin \\gamma_{1}}{\\sin \\gamma_{2}}=1$.\n\n因此, 由式(1)知\n\n$A L 、 B M 、 C N$ 三线共点\n\n$\\Leftrightarrow A D 、 B E 、 C F$ 三线共点.']			['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ApVSowsKggFgMn7q8fe9R8DeB9O8C6GmnWBeWV8Pc3L5Bmk7nGSFHYAdB3JyaTwL4F03wJowsLECW4kO+6u2QBp3HQnk7QM4C5wOe5JPV0AFFFFABVLUtStNI0241C+mWG1tozLLIQflUDPQck+wq4wypHtXjPxh8VXOoajafDnRBE97qrRx3UzMT5AZ1Kr8pJGQNzZH3CODngAzfhXaz/ETx7qvjrXImmhtJBHp8Mw3pA2dyhTwP3a4/h5aQPw3Ne81heFfDOneEdAttI02ILHFgySY+aaQj5pGPOScfgMAYAFbtABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAeRfEz4cXUssXirwTAbPxDBK0s32V/Le4DDDMB90vySem4M2cnArY+G3xHtvE1lBpGrT/ZvFFsWhurSdfKeV0zuZRwCcAkqBlSG4AANeiHoa8a+Kfw4vF1SPxz4QEset20iSz28KFzMwIxIi4Pzf3l6MPfO4A9morzj4X/FC28c2osrsLDrsMbSzwxxMsTIHwGQkt2ZAdxByTgYr0egAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigApD0NLRQB59ffF/wzp+uPos8eqDUFlEIgFi+92J+Xap+Y7sgjjkEYo1P4u+H9FEX9q2WuWHm7vL+1adJFuxjO3cBnGR+YryDRUPjT9o97mZ2eKDUHmEkYCEpbg+VkEd/LjBAGSCenWvd9ZSz8a+GPEmi2UlvcSoJbFjLho0uPLVl5AP3GZecZDD1FAGn4d1+w8T6FbaxprObS43+WXXaTtYqePqDXh91JceK/2noEi/0q20qVFH7sFYEiXc24gHpKWwT/ABEDjjHtljDb+EvB9tDdXWbXSNPVZrjyyMpFGMttGT0UnHPpzXj3wQNsZPEvjzXbuyt3vLprcTyzeUFZj5soIYhQCWjx1PynoOoB73WB4k8XaJ4QsRd61fpbq4PlR4y8pGOFUcnGRzjAyMmvJfEPxt1jV9R+wfD/AEi4vFj3CW4e0eV3OTgpGp4UqpOWGTnopXJm8N/A+81S/fW/H+ote3Vxh3tUkZjnC48yXI6Dcu1OOBhsDFAFfXPid4o8f3Z0T4c6feRQAlbi/KgNtZTj5jkQjhiDneSBjaRg6HgX4HR2t1JqvjZotTvMoYbfzWeNMYO6QnG9uNu05XGfvZG31nR9H07QdOg0/S7OO0tYgNsUQx2xlj1ZvUnJPc5rSoArWlpbWFrHa2cEVvbxLtjiiQKij0AGBXzr4f0WL4tfFLxfq0hQ6esEsVvcJGCisyeTA7RudxOxWfjBDIOV6V7P8SdXl0P4d65fwzNBMtsUjmXcCjuRGCpXkEFhg9jiuM/Z60maz8C3OoTRsn2+7ZovmUho0ATOByDvEg59B7EgGX+z3NY6p4O1zQrq2W4Ed2Jpo5o1aJ0kQKowc55ibjHp17Z2s6Pr3wW8VN4g0BJ5/B88q/abUyF1jDcFWGeCCMLJz1UEnJDO+Ct3dWHxY8WaLLsRH855EOCfMin2gA/R3+vWvfLy1gvrKe0uollt542iljbo6sMEH6g0AZnhzxJpXirR49U0e6W4tnJVsDDRuOqsvUHp+BBGQQa8Y8TX63f7T2jwCBYms2ggZwcmYlDJuPAxxIF6n7o+gr67Yan8CPEcWr6DILzQNTZY5rW5b5gy87CwA5wWKvjuQwOMtj2+t2fiP9ozTdX092a1up7Z03Y3L/o6AqQCcEEEHk9DQB9QUHoaZLIkUTySOqIilmZjgADqSTXg/if4max4+12Hwf8AD8zW/myMJtRMhiYhWJ3IyklYsDcT95shQB0YA1viT8XJrHVP+ET8JIl1q8j/AGaafkiKRhtCR4IzIGYdeFIxgnO3V+GPwqi8KwvquurBfa9O/mb2AkFsck/IxGS5zlm/AcDLaXw5+HOn+ANI82Typ9YmT/TLwdAOTsTP3UHGTxuwCQOAvC+OPG2q/EPWz4H8CFpLV8x39+h/duv8XzjpEOQSPv52gEEbgA8b+PdX8f60fBHgLdJG24Xl/G+1XUHDYYH5YvVur5AHBG/03wN4F0rwLo62VigluXwbq8dAJJ2/oo7KOB7kklvgnwBovgjTYoLG3jkvym24v2jHmyk43c9VTKjCg4GM8nJPWHO04IBx1NAC0V5/8TfFHibwjo7axo8eitYwKon+3NIZpHZwoVEXA4Byct68DHNP4SfETUfH9nqjalBZxXFm8YVbWN1GxwcE7mPdT37UAemUV5p4z8T+NdJ8a6ToXh6LQ7uPVVYwrdJL5sGwDe0hEgGzByCBk4IwSBu9BsPtn9m239oeR9u8pftHkZ8rzMfNtzztznGecUAU9f16w8N6HcaxqczR2duql2VCx5IAwBzySB6V5N8KPDV14m8W3vxM1aONIruWU6dB5m9kO4puJGBhVBjAI55OBgEx/HrUJNZuNB8F6YoutSnuBcPCpwykgpGCTwM7nJ5GAATgEGvXfDmjR+HfDmm6PE/mLZ26QmTBXewHLYycZOTjPGaANaiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACkPQ0tFAHkXxR+F99rV9beJPCTJaa3BnzVifyGn6kOHBGJBkjJIyCPmG0Z2Pht8ULHxjHHpd4r2viGCLNxBImFkK4DMhx+JU4IyeoGa9EP3T9K8a+KvgR9LYeOfB9vcW2vW9yJrj7JyHUghn8vByckbuxUuWByaAPZqK4T4XeO08b+GI5rqe3GrwsyXUEXy4wflYKSTtKkc9M7hxiu7oAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAqjrWpJo2hahqkiGRLK2kuGQHBYIpbH6VePQ15d8bpdYuvBn9h6Po19qD30qCdoLZ5RFGh3jlejFlUcgjAOcZBIB4N4Q8CX3jfTfEV/bXRNxpcImEPlNJJdOwdggwc5Ow+uSR9a97+A9itl8L7acbf8ATbmackE5OG8vn0P7vtx07mo/ghE+meEv7HudB1bTb+KRri5lvbV447h3YgFC3oixgjjkdDya9TPSgDyn4+6ymnfDiTTyheTU544V5xsCMJC3Tn7gGOPvZ5xXy5aG2F7AbxZWtRIvnLCwDlM/MFJBAOM4yCK99+OlyniDxX4a8KWkxa4EpM6RyREhpCgQYZ1+fAJAYqDvXnnjB+I/gnTvh/430nW106OfwvcSxpJaLGz+WEUK6EscFmXc6ksCSG/u5IB2fw/+KHw30+yGmWNqfDpaQALcKWWVmY8tMNx44yZMBQcA4FewW1xDd20NxbzJNBKgkjlQ5V1IyGB6EEHP415B4j+B3hvxJpceo+FJxp008YniOWkgnVl3LwTlASVwRwBn5T28pk+HXxF8H6q93p+m34mgJ2XWmMZN67iMjZ82D12kdOoxQB9eUHpXyTo/xv8AG+ksxk1CLUoypAivYQwBODu3LtbPXgnHJ4r07w/+0PomoXbQ65ps+kxnGy4SQ3CDgk7sKGXoAMBs57YoAn/aD8RnTPBtvoaJuk1eX5mZflWOIq5wc8MWMeODxu6cV6R4Y0dfDvhjTNIjEX+h2yRMY02K7gDc2P8AabLfie9eDXfiCx8eftFaHcWs9vLp1s0K2sxDx7xGpm53YO7zCygYAOF+p+kj0oA+fNDtodE/anvrODzWW7eWTc+Mq8kPnt26ZLAe2Oe9fQZ6GvDfidLe6H8bPBviAzRWliwitWuZHjAx5ribIbkARyjLYwN3BzXt2R3xgjpmgCpqulWOtaXcabqVqlzZ3ClZYXHDDr16gg8gjkEZHNfJt9b2/wALvi+Y08y9ttLuklUNhXZGRWAJHG4B8ehI6DPH1pqOpWek2E1/qF1Fa2sPLzSuFVecdT78Ad818n+IpLb4l/GWYabOIbbUrqOGKd0J+RUVN4XgnIQkA464OOTQB13i/wAaal8ZNS0/wt4OgvYbNh5t4tyFQNgjDyFS37teD15YjgkLXsfgXwPp3gXQlsLNPMuZMNd3RA3TuB+ijJ2r2z3JYnxC5tPFfwG8QRXFtOdR0G6ZPMJjKRznByjfe8uQYJBBOQAeQCtdJ8QPiN/wnVvD4Q8AfatQub7Jup4g0I8sAgx/OF4PVicLjjnccACeNfHGqfEPXj4D8Dt5lrIxW+1CJvlkj4D4btECSCed5wBwQH9O8DeB9L8C6KLGwHmXEuGurtl2vcOO59FGTtUcAHuSSV8EeCNM8C6Kmn2AElw+GursoA9w3qfRRkhV5wO5OSepoAKQ9DS0HpQB5Z4/vINc+JHg7wVNaRTQNN/alyJ0yrKiyBE68/dkyCMH5ecZrzvwmY/h3+0LdaN+4isrqV7KPbvk2Ry7ZIFB67s+UpJyOW7fMPRfBgXWfjJ421l7iRzYJDplvGJw6InVxjqPniz2GWfqea4P9oTT207xPouu2ca2tzNGQ1zC4SRpYyCrcHOVBX5u3yjPAoA6/wAN3sviz4+63qSQFLLQLQ6arcKfM8wj5hkk5PnkEY4C5wevo3ibxBaeGPDd7rd6WMFrHuKqDlySAqj03MQMngZrjPgppk8PgybWb8O2pa1dyXtxLJbeU5B+7z/Ep5cEYH7w4Hc8L8afE02q+OdP8FDUBZ6WrQLft5nlqXkZTlySRsVdjDIwCSTnjABo/A7QtR1vWNQ+IGuytdXNwzQW8k8fzM3G+RcjAAA8sbeMb14Ar3Wqmn2Nvplha2Fonl2ttEkMKbi21FACjJ5OAO5NW6ACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAo7UUUAeH/E34falpGuN8QvCMnl3luyTz2cETb3fO1nUJ94MG+cYGRuJJya7r4ffEfTPH9nK9rFJaX1tjz7SRt20HoysMbl4IzjI7gAjPbHgGvB/ib4Ov/Bd7b+MfAFtNp7IjjUY7IZjRRhg3lYxs4O4Y2japwOTQB7xRXLeBPGNl4z8OW17BcwNepFGL2CM4MMpHI2kkhSQ20nOQOvWupoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoPSis3xDfnSvDWq6ipIa0s5pwVAJyiFuM8du9AHjfgC+fxP8AH3xDq5mW8s7KCaC1mmdZtiCRUUxEABQRvIIB4YglixY+teLPDkPi3wtf6HcStCl1GFEgGSjghlbGRkBgDjIz04rzL9nfw7PYeG7/AF2f7upyqkCkDOyIsC2c92ZlwQD8meQRXtJ6GgDw34LeKbzRNSm+HPiGGS1vreRzZRtGMg/NJIhYcEYJdW5BBPzfcFe5dq8N+OXh/UrDUdO8eaKFhl08KtzOpUMh8wCJsE4YEsV6HjGcjp634a1+08T+HrLWrJiYLpMgEEbWBIZeQM4YEZ744oAyPFPw68NeMVMmp6ehvBE0aXcbMkiZGASQfmIxkB9wHPqa8c8S/s8anY2zXHh/UF1JwR/okyLE+OBw+dpPU4OOB3PFfSJ6VynxF1k6D8PNc1BXnSRbVoo5IDhkkk/dowORjDMpyDkelAHxcRyfbvmvQ/BHj34gaa8VtoL3WrwRKcae8T3YRFUKPlHzog3LjaQMjvjFevfBvwbp/wDwrKCTVbGyvf7Rme62XNsj7EyFUEnO7PlhugxuxjiqHi74BWF20N54PuBpNzEM+RNJI0btuGGDkl0YDPZgcLwvJIB5Z48+Kd94+0i1sr3TorY283mh4Jn2t8pBBQnB65BPIGQDyc+xeG/jX4Li8K2Ed5eS2d3b2yRvbG0P3lUA7fLXywpIOB8v0XpXi/jjSfGnhezg0rxPN5tpcSmS3l81JjIUVVwJDmQKoZflOB6DvXKaPa6fearBb6pqLadZuSJboW5mMXHB2AgkZwODx6UAetQXXiH4/eJvsc8i6VoGnDzZIosuFJJAz03yHkAnAChiBkkMa94XsPBXxt8Gab4dhlSNltHlZmMjSkzOjuc8AlBzgADnAFVrb4VeKrE3N94C8WWmo2h/dmbTNQMLuQoYo207c5PALehOM1554q8Q614k1j7Rr98l3e26fZvNjEW3arMcKYxtYZLHcMg564xQB658VPH1x411CDwP4NaLUobtlFw8CEmWRXJ2KzDbsXYrlxwR3ADZwNKk1j4EeOj/AGvYx3lhfRhDcQqMyxZBLROwyGX+JCQDxnjaw1fhB478C+DtFaDUHubbVrl2NzdPb+YmATsVCmW24CnGM7ie2Me0x3fhX4g6LNapPp+sWUkamWFXDtGHB2lgPmjbrjowIPQigC54b8Raf4s0K11jSpWe2nGcOMOjDhlYdiDx3HoSCDWzXzRfWXiL4E+MEvbOW4u/DF1Ntw/KzKeSjgcLKAPlbjOCem5a9/8ADfiTSvFWjxapo90txbOxVuzRuOqsvVT0/AgjIINAGxVS/wDtosZvsH2c3YQmITg+Wzdg2OQPcZx6HpVuigDz7wBpHjTSr7V5PEX9jrb6hey3jC2Z2lEhwgAH3QhVFYEksB15J26nxC8If8Jr4RudJWcQzbvNgdkUgSKDtDEqSq88lcNjI5BIPW0h6GgDidQv4Phf8LY5JS9y2mWkcMavIW86U4VQCxzt3HOB91QcDAAryTwb8LZ/idpF34s8Q63dreX8ziJ1jBztIXc2eq5BUKu0AKOew0fiFq158TviBbeANEuAmn2cxN5M8WMSoSsjcnLqgOAOMtu6jaw9003TrXSdOttOsYFgtbaNYoo1JIVQOBk8n6nJPegDwr/hFfib8LJFl8O3jeINGjBAtNrOFzjP+j7iw+dyf3ZP3ctgcVs+EP2gdM1W+jsfEFkNLklcIl0ku+DJz9/OCg+6M/MOcnaBmvaT0rn/ABD4O0HxRaXEOpaZbySzRsguvJTzo8jGUYgkEcY7cCgDVsb601O0ju7G6gurZ8hJYJBIhwSDhgSDggj8DVuvC9X+BeqaOBdeBfE17bSgfNDcTmJ2OGyVkjA7bQFKjqTuHSm2XxJ8e+A71YviJo81zpsuSt7bxR71Yg7VVkIjb7jHYcPgk5wACAe7UVw/hv4r+D/E1zDa2mqCC8lxstrsGJixIAUH7rNlsAAkk5xnrXcUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFB6GiigDwXxf4K1D4Za+fH3hFg1ksjG+08DascbnkADAaPP8OPkIUgEDK+oeBvHGl+OtFW/sD5dzHhLu1Zsvbuf5qcHDDg47EEDqJI0lieORFdHUqysMhgeoI7ivAPiX8P9S8IaunjDwHA1jb20Zku0s2OY237i+wkjy8EZUDaApyMZoA+gaK5XwL440vx1oovrB/LniIW6tGbLwOc9em5TjKtgZx2IIHVUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAAehrxz9oPW5rTwrp+hW4lM2rXPzKqBhJHGVJX1B3tGRgc4PPY+xnpXgHiq0n8W/tKabp0cOY9LW3adjEWURp+/O7HY7wmTjlgPqAe0eG9Gi8OeHdO0aHYUs4EhLpGEEjAfM+0dCxyx9yeT1rXPQ00ADFOoAoarpVjrel3Gm6lapc2dwpWWFxww69eoIPII5BGRzXh/we1K+8KfETWPAF2dtp500kKyph/NTGGBO0lXiXPQ8BSMDdXv9eM/HXw7PFbad4502VYb/R5Iw7Ebjt8wGNgGJHyyMeMc7+ThaAPZu1eIftH3rnRNB0mNGdrq7eYKq5JKKFAH/f3pXo3gLxX/AMJl4OsdZMYjuJAY7iNAdqyqcNjPY9RyeCMnINeYjyvHP7S3/LvLY+Hocj76MxhP6ss8nspC9+4B654V0dtA8KaVpMmzzbS2jilMbEqXCjcQTjgtnHHpwOlYPxD+JOm/D+xhM8ZvNRuDmGzR9hKAjLu2DtHbocngDgkR/Ej4k2HgLTAq7bnWLhCbW16gdg8mCCEyPq2CB0JHAfDX4bSeL5m8beOBdXdxdTie2gnwFuF28O690PG1flGEHBUgUAZemfDbxb8UTdeKvFF39mNxbOLGJzsbcB+6+TaQkOST/ePJwd241/g54I8KeONA1e21awnbUrSdD9pjmZNsTr8oAyQSCj9V6Ecnt9KAfTpxXx5FLqg8W6/4S8F3DCw1i+e1WO2O9ZYEkcIfM5bYFJyQcFSc5FAEmq6PJaeOL7w38PtQ1e/gmH2WUISpkPSRWZSFePJOWYKOT2+Y4Fkb7wf4r0681HTbmK4sbmG6NrcKYWcKwYDkcA464PrzX1R8N/hzZeANMkQS/atTucfabojAIGcIg7KPzJ56YC+a/FuCG6+Png+3uIkmglSySSORQyupunBBB4II7UAbdlbfB34iWkcsVrY6XdDKCDctlMuDnO1W2PkDqN2AeoNQ6p8A10947/wZ4hvLDUodxX7TIRw3HyyRqGT5S46NnOOOTWp4q+AnhzW5BPo8p0W5aQvIIkMsLg5JxGWG3kjG0hQBjb0xxyaZ8UvhDbzS2Ri1bQooyzJlpoIhliW8vIePGSzFfl5ySccAGd4guvitoOiXXhrXNPm1fTHVg1zLbm7V0LkA+aOR83K7sOvynjgDjfBHjvV/h7rMstrGskUpWO7tJcjzApPf+FhlgDg4JOQele1aD+0P4ev5ki1ixu9LLZzMp+0RJ1PJUBv7o4U8nsOa77UfD3hHx1YC6u7HTdVt5ECJeRFWbarE7UmQ7gA2QQGHOQepoAn0HxjoPiLT7S9sdStT9oACwNMglRjg7GUMcMMgY59uMZ6Cvj34kfDa+8BaiHG+40a4ci1uiOR6RyYAw+BnoAwBI7gdR4b/AOFqaH4U07xB4d1GTWtHnRP9DjLXBhWNtgi8uRd4GSynyv7uc4CmgD6aPSuS8f8AjO18EeGJ9RmdTdyKYrKEru8yYgldwyPlHUnIwM4OSAfNrD9oU2chsvE/hm5tLyHCz+Q2CG7/ALpwCvbgsa5vxR4k0j4xfEXw1pUEt3p+mjMUkl2yIxZjuZUALLuYKqj1YgY4GQDvvgp4Hm0jT5fFWrKRq+rAunz8JbuVcZUYAZm+Y9cAKPlO4V65UUUaxRpHGqoigKqqMAAdh6CpaACiiigAqO4ghuraW3uIo5oZUKSRyKGV1IwQQeCCOMVJRQB5dr/wJ8H6x9olsoZ9KuZdzq1s2Y1c5IPltkbckfKu3gYGK4y5t/i18LYvIsZjruixtmI+WbgIgyArL/rIwFQZAOxc8HNfQlFAHk3hX44aHqktvp3iCOXRdV24nedQttvHGAxO5c/e+YADGNx4z6dZX9nqMC3FldQXUDdJIZA6noeCCfUH8RWB4s8BeHvGtqY9Wsg0wAWO8iws8YGcAPjlfmPynK5OcZ5ry2b4L+LvClzcXvgjxS4Xf5otndoXk2liiNjKSHoPmCqdxyAKAPfKK8Hs/jdrnhi5XSPHnhudbyLANxCPLeRR8u/YflfJDnerBT0AA5r1Pwv428P+L4I5dI1KKScx+Y9o7BZohxncnXgkDcMrk8E8UAdLRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFB6GiigDwTxj4Ov/hVqo8beCD/AKJzHeWUiF1jVjnPHPlkgZ5ypAwSD8vqngjxnYeOPD0OqWQ8t87Li2LAtA/cHHUeh7gjgcgdOelfO3jzw1qvwu8Yx+NvCsAi0d3UXMEJOyMsRujdccRucY7KxGNpCUAfRNFc14L8Zaf430BNW05ZY18wwzQyjBikABK56HhlII45HQ5A6WgAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigCvf3tvpunXN/dyeVbW0TzSvgnaigljgcnAB6V4j8ArC61TWPEXjK/iImu5TEkqkBWZ2Mkw2/XyjnpzgHg11Px2v0svhddwSDLXtxDAn1DCT09Iz3/ABrR+EGjSaL8NNIjkcNJdR/bCVOQBJ8yjoOdpXPvnnpQB3lFB6GvIPi9q3j3wxC2u6NrEcGkCSOA28Vksjx5BJkkZ1bjdhew5UYzkkA9fqvf2VvqWnXVhdx+bbXMTwzR5I3IwIYZHPIJ6VwPwe13X/EfhNtU1zV7XUDLMwjEcSpJAVJDJJtAXkBGHAOG6kEYzdL8aeMvE/jXXtI8Pt4ek0zS5CpvZoZsHJIVMCQFj8rgsBtOwkdVyAcZ4c8TN8HPGHiDwlqM3/EnO+4spDCZCJSgMZcqFJDKFVsAjcvGBuNcH4O8fz+DjrepxQm716/VI4bm4O9UUszSu2fmZiQmBwDyTnAB9f8Aj34Hk1jSE8U2S7rnTYilzGFZmkg3ZBGMgbCzMePuliW+UCvIvAfhfx1dzp4k8G2e6S0leFbjzIBscp8w2yED7r+mOaAPU/h98LZPEMv/AAmXjppr6/vZBPFaT5UYHRpV4znAwgwoUAHOdo9vAHHTgda8C/4yG/z/AGfXFX3xR+Jv9o3OgzaxOb7zWs3ht7WHzPMyUKoUTO7PAKnOelAHb/Ez4ian4o1ibwD4Ot5pmdzbXUsQIaZgcNGp42xjkMx4Iz0XJbmvCmgH4dfHbQtN1S9ikYgAyxKxHmTQsgXGM/6xtoPcYJxk4qaH8Lfifout2V1p9g+nXHmqq3S3kRVO5L7WJKYByMEHpg5ANb4ieHPG2i+LtPv9f1GK61S/Km1vLKUR4eMqoA4TYVyhzgD5s5JzQB9bnpXgXxO8v/hoXwZ5jMBiyxtXPP2p8DqOM9/0qrdfDb4xW4g8rxfcXRklVG8vWLgeUp6u24D5R3xlvQGuD8Q+HfGVj8QtL0bW9Ulk1uaSFbK9lvHk2hpCqMr8uqh8noCCCcc0AfYdIeQRXgH/AAq34vf9D5/5WLv/AOIqtf8Aw4+MllbebB4rub9wQPJttZnDH3/ebV/WgD1rxL8N/C3i6X7TqumIbs9bqAmKRuFHzMMb+FAG4HA6Yryy++C/izwhcNqngTxBNM/mbjbF/s8pUMCikk7JcAktu2g46HOKWH4a/GC5toJJPGjQN5YJifVbjehPJDFFIYgnGcnpgEgCqlp8PPjLc3txBL4mvbVIiQtxNrMpSbBxldhZvcblHHXB4oAqzfFvX1s28P8AxC8Nm40+5i8m4LW7W1xyVPmAN8pZQcgBV52ncK5XwJ8S774ealdx2SvqGjzOS1tOfKLEZCuuCwjcjbuHzZAxk4BHX3Hw0+LevadLY6prTNawwpiC61JmWcnMhUhc7mVjjc/oNpKgEc3N8DfHkdrbyppkEsku7fBHdRb4dpwN2SFOeo2lvfB4oA9l+Jt94WvPhneeJvsul38l1afZ7C7mgRpCZCQAhYBgV3O2ByCrHHBrgPAXwPs/E3gdNX1e8vbK9u972YjxsSPACM6MoJyQW4YAqVwRmuE8E+C7jW/iTa+HL+1YC2uWGoIMsESI/vFZkPGSNm4HGWGOtfWWpeJdC0e5S31PWtOsJmUSCO6u44mK8jOGOSMgjPTg+lAHjKeEfjH4LkWHw/rK6tY52JG0yERxJ9wFJ8eWCD92MkDGM8Amk/xo8eeE9UFp4q0CBkDuCkkTQPIB8vyOCVKg4OQGyO/INe1/8J34Q/6GvQ//AAYw/wDxValhqdhqtr9p029t7y3JIEttKJFJHXlSaAPN/D3x38I6xJDBevc6TcSBMm6QGHzG4KiRScAE/eYKMdcdB6LYaxpeqBTp+o2l2Cu4GCZZPl9eCeK53xD8M/CPiWOcXui20dxKXkN1aqIZt7dWLKBvOTu+bIz1BrzDUv2dry0vhd+G/EflvG/mQC5Uo8RBypEid89woxQB9AUV8+T638Yfhvby299bJrtgiGQXrxvdLGqjc7F1KuB82P3v935eAa2NJ/aL0O6mij1XSLywVztaWN1nRDnqeFO0Dk4BPYA0Ae10Vg+H/GHh7xOqtousWt2xVn8lX2yqobaS0bYdRn1A6j1Fb1ABR2oooAzNb0LTPEOntYatYwXls2TsmXIUkEbgeqtgkBhyM8V5fr/wD0iYrfeFb250fUISJIVZ2eIMoJXk/OjFtp3ZOMfdr2OigD56bxl8WvAN1HZa5pR1izjZVWdoTL5sYJXCzR/xEKTlwX5ywPSu08O/HXwfrAiivpZ9JuW8tdt0u6Pc3XEi5AUHqzhOCD649Qb7p+lcn4p+HvhnxfFKNT02JbqVgxvbdVjuMhdoO8D5sDAw2RwOOBQB0Vlf2eowLcWV1BdQN0khkDqeh4IJ9QfxFW68Bm+DPjDwpfTXngfxOwiz53kPI0DyFCSkbYykp7fNtXk5ABq5a/HXUNAni03xt4WvLS6WMl5ofkMnzEBlifHynGNwcjIOPQAHuVFc/wCHvGfh3xUP+JLq9vduAXaEHbKACASUbDAZI5xjkc10FABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABUN5awX1lPZ3UQlt542iljbo6sMEH6g1NRQB89a1per/AvXxrfh4NfeG78qlxbT5yhU52s4HHBbY3bJBBx83t3h3X7HxNottq2my+ZbTqCMkbkPdWAJwwPB5P5Vfv7G31PTrqwu4/MtrqJ4ZkyRuRgQwyCCOCenNfOFzp2u/AfxpHqNt5154avXMZwwxKvJCScYWVRkggDODjguAAfS9FZmg6zaeINFs9WsH32t5EsqZIJXPVWwThgcgjJwQR2rToAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACg9KKD0POKAPBfizeDxl8TfDngSA3DQwzq9+IJCv+s2sflIxlIgWDc/fI4wRXudvbw20EUFvEkMESBI40XaqKBgAAcAAY47V4D8ORN4q+P+va/cJM6WRmaKQphU58mNGK8Z8rdjPXYTzya+haACuO+KNmb34Y+IIREH22hm2k4x5ZD5H025rsT0rn/GurS6F4J1rU7Z2juLezkaFwoO2TBCHB4IDEHmgD5w+H/j3VdE8Kaj4X0Cwnu9b1O6JtDEoIhBTDuMclgFBH8I+8TgYP0n4V8PWnhfQLbTrWMBwA9xLks08xHzyMx5Zie57ADgAAeLfs36IjXus66zv5kaLaRqDgEMdzEjHP3Exg/3vavoWgBk0STwSQyoHjkUqynoQRgivnnwbPB8IPitqnh/WLgwaVqMa/ZbqQgqRuPku7YAAwXViBgN/sgmvok9K+c/2i9c0+51TSdGjTff2SvNNIHBWMSYwhAOd3yhuQOCpH3qAOk+JnxMup9RHgjwUXl1qeT7PPcxPs8ls/6uNsgB+OWzhRkfezt6L4cfCnS/BFvDeXAS81xlzJckArCSCCsWRlRgkFurc9AdtYfwD8N6NbeFI/EcBWbV7lpYJ3EpPkpvXERUHA4RH5GfmHbFexUAIRlSPUV4n+0fp0MvhjRtTdnE9vemBACNhWRCzZ75BiXHOOT+Htp6VwPxh0p9W+F2sLHbRzzW6Lcpu25jCMGdwT0IQP05wSB1xQB12k6hFrGjWOp2yyLb3lvHcRrIAGCuoYbsE84ODz+Jrxzx/DHcftHeDo5F3KLe3cc91mmYH8CBXdfCTWf7a+GWiSl4WltojaSLEfueUdqhhnhigRiP9rIGCK4jx1/ycl4O/wCvSL/0ZNQB7hRRRQAUUUUAB6GvOfi/45n8F+Eh9hMY1PUHaC3LOQYl2nfIoHUrlQOmCwPOMH0Y9DXzqJ5fjh8UEgk8tfDGil3UDIaaPcBnOA26QqvHG1R/eHzAHf8AwV8KxeHvANteeW323VlF1O5Ofk58oDgcbTn6s3PSoLXwAnin4h+INf8AF+lyvbxyLa6Zazz74jEqj94MHv8Ae2nABd85PI9PA4Ht714V8efH9zZD/hENMm8ozRCS/likG7Y2cQkdVyACehIKjoxyAeZ+KLWw8Z+Pv7O8BaGRbqnkxpDwJypJaU54QHIAJ7AE4JwPqTwd4YtvB3hmz0S1laYQAl5mQKZHY5ZsDpyeASSAACT1rivhD8Lo/CFkms6mPM1y5jxs7WiH+Aern+I9ug6Et6rQAUUUUAB6Vzmt+B/DHiJ5pNV0KxuJ5SN85iCytxtGZFw3THfjA9K6OigDxPxN8AbV7ldR8G6jJpV5E6yRwTyMyKwK4KSDLoRhmyd2SRjaKycfG7wZfbvn1y0iULyRdRy4T/gMvGf9kkgfe7/QVB6UAeKaH8fbWO4/s7xjo1xpN7EwSSSNGKq3OS6H50xxx85Oa9F0Hx74W8SzpBpGt2txO5wsLMY5WxknCOAx4BJ44AzV3XvDGieJrYW+taXbXiBSEMq/NGDjO1h8y5wPukdK8v1T9nXw/czSSafql/ZbyxWJwsqoTkgDocD3JPHWgD2mivBptM+MHw7jD6dfr4k0uORgICrTybeETKHEo4IO2NmA2nPHVumftEzWt2LPxJ4beGSNmS4e1fDxsBgr5Tgc7hggsMe+OQD3uiuR8L/Ejwr4ulSDS9UT7Y6qxtJwY5ckEkBW++QFOdhYDGc8jPXUAHas7V9G0zXLP7Jqthb3tvu3COeMOAcEbhnocEjI5Ga0aKAPG/E3wF0+6vl1PwrqD6LeR5kSHDNH5o5Qq2Q0fzdxuxxgDGDzFp4z+Knw+eS68U6VdajpS/60zlSEJACkToGA5K9dw6gYPNfRdB6UAee+GvjD4P1+xgebVINMvXX97a3j7PKbnIEhAVuhIIPORkAnFd+DyOnIrzXxJ8D/AAh4hn+020EukzlTn7BhI2OFAzGQQMY/h25JJOTXFjw38UPhW5/4Ry4bX9EQA/ZvLLgZJyBBnevzOT+7Y5wC3HAAPoGivF/DX7QOlXNzLaeJ7GTSJY+POjDSoWGchlC7lI4HGcnOcV67Y31pqdtHeWN3Bd2smdk0EgkRsEg4YcHByOvUUAW6KKKACiiigAooooAKKKKACiiigArP1bSbDXtLuNN1O2W4sp12yROSAec9uQQQCCORWhR2oA+bLXUtd+BHjGSwuopbrwrfTvJCu9WaROB5inAxKo2hgQAen91h9C6ZqdlrOnwajp9zHc2c67opY2yCM4P45BBB5GCDjBqLWtC0zxDp7WGrWMF5bMSdkq52kgjcp6q2CQGHIzxXhGm+Ite+CHiE+H9dWW+8LzO8tpLHGGbbg/6vLKFO4rvQ9DyPvZYA+iqKqaffQalp9rfWkhktbmFJoZMEb0YBlODyMg9wKt0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFYnirxDb+F/C9/rdyN8drFuVDkeY5wEXIBxuYqM4wM1tHoa8O/aM1y5tdI0jRIZVWK9eSa5RWO4iPbsU8jKksx5HJQEdKANX9n7RPsXgq41iWP8A0jU7pmEu8ktFHlVBHQEP5vbJz9K9drK8NWf9neF9IscSD7NZQw4kGG+VAOffitWgBD0OK81+JPhbx34rEmm6Pq+k22iTRgTRSq6SyHjKswV8jK5BXbwxBBHJ9LooA8j+FXw88X+BdUmW+1TSn0idS0sEAeR3k42kMVXbjnnJHbbk7h64elIehryv4m/E6TRNnh/wo32zxJcPt2QRecbcDPG3nLnHC4OBkkDjIAfEz4n/ANhzN4W0CKS48S3eyFNvS3MnC98mTDDaOmSCTxg+TxXi/Dhp9O01rfWPG9/Iba6cQm5jswwIMUef9ZOWZd3BXI24POYYL2bwzeQWeir/AGp49vJHF1qCyi5EBlXmOIgkNL83zyfwkMoLAk17T8OvhHpngh4tSuJDe62YdjynHlQsfveUMZBwdu48kZ+6GK0AeT/DeTWfhn8VrTQ9age2TU1jgmiyXVi/+qZdp2sQ52buQMuK+oq82+M3g9PFPgqW8hhkk1PS0ae1CtjKnaZVI6NlVyB1yoweoLPgl4rPiTwLBaTBFutJK2ZCk/PEFHlt044yvU52E9+AD0ysXxhbPeeCdetY2VXm064jUt0BMbAZ9ua2qQ9DQB4p+zjqMU3hfWNMUSCe3vFuGbA2lZECqAeucxN+BH4RePHSP9o/wezuqr9mhGScDJlmAH1JNUvgHbXOi+MfFmiXDxtJblY5jHyrPFIyZXIHHJ7dx0q946/5OS8Hf9ekX/oyagD3CiiigAoPQ84opskiRRPJIyoiAszMcAAdST2oA85+MfjOLwp4MntI2I1HVY3t7YbWIC8CRywIxhW45J3FeCM4tfCjwO3gjwgkN5Ei6rdt594VZWKnose4AZCr2yQGZiCQa8z8N6ZJ8V/jJf8AiKeadtD0ydWgfGVkEbDyowGwVDYMjDbxkg4LA19D8KOlAHDfFDx1b+CfDTslw8erXkbpYKIt/wA4xl+RtG3cOG6+h5FeP/B608NX+qX3ibxprOnSXccoW2i1K9TczkZaV1c5bGVCkk87u4Uj3bxH4C8M+LriCfXNKS6mhQoknmPGwUnO0lCCRnJAPQk46msX/hSfw8H/ADL3/k5cf/HKAOdl8ex+Nvip4f0Dw5qd0uk2kkl1fXFtJ5YuWQEqhyMtHkAEdGDnj5Qa9irk/Dvw68KeFdT/ALQ0bSFtrsxmLzDPLIQpwSMMxA6dfw711lABRRRQAUUUUAFFFFABRRRQAVUvNOsb9Ql5Z29yqggLNGrgA9Rg+tW6KAPO9Z+C/gnWbk3H9mNZSMCGFi/lIcptBCcqpHDDAGSPmyCQfP7/AOB3i3Q76SXwj4m22o3SIGuJLaZSSRj5PlPy4G7K554HFfQlFAHgcnj74qeA/n8W6Cmp6erlpLpUC4U/Ko82IFEG7BAZMnOO4x1Gl/HvwVfqPtcl7p0hHzCeAuucDvHuyO3I7dBXqh6Vw2t/CfwXr0zXFzocMFwygeZasYO5JJVCFLHJyxBP5CgDrNO1Kx1a2Fzp17bXluWIEtvKHXI6jIJ5q7Xid58CrzRr19S8DeJ7vTboKFEU7EBvnBIMic7MAHaVbJXnrxj3HjD41eErdZtY0ZL6KTOGe1SUJjb1Nuw28njd1ycZxwAfQlB6V5R4d+O/hrVLqKy1iC50W9YlW+0YeFX37Qu8YIPQksqgYOT6+i6brukawW/szVLG92fe+y3CyYxjrtJ9R+YoApa74P8AD3iWF4tY0e1uiwA8xk2yjBB4kXDD7ozg8gYPFeQav8F/E/h7UBc+BdcmW1jUOsU14Y5jL0bG1QhBAHUjPIPQV79R2oA8Q0z4ya34Zmj0/wCIvh29tpMtFHfQwbRKU2qTtOFYZyxdGxyuFxg16F4Z+I3hTxbOttpOrRtdkA/ZZlMUh4LYVW+9gKSducD0re1PSdO1i3S31PT7W+hV/MWO5hWVQ2CM4bjOCRn3rzbxD8BPCuryXFxYyXGlXEpZ1FvhoVY/9Mz0Xr8qkDnjAAAAPWKK+eZtX+Mnw/m+xzQPrtnGhdZ/Ie7THJbMi4k4z/HjG3jjr1Phv4+aDqt+tjrdlPoly0pjDSOJIV6Ab3wpU5yDlcADJI7AHrtFV7O8tb+3S5s7mG5gf7skMgdT9CCRVigAooooAKKKKACiiigAPSsDxX4V03xjoEuj6oJBBI6uskW3zI2U5yhYEKcZGcdGI71v0h6GgDwDStf1L4JeKpPDutpNP4UvJXlsZ1+dogW+8DhQf4d6ADBO5evz+8208V1bxXFvKk0Eqh45Y2DK6kcMCOCCO44NZfiTw1pfi3RpdK1i3863kIZSDho27Op7MOfwJByCRXhnhfxTrPwg8TzeGPEy3T+HfOcW9y0RxGGORMmM7kbqyBjgkn7wIIB9HUVFFKsqI8bh0YZVgcgj1qWgAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAA9DXgfiW1j+IX7QtjpkBSaw0SJDeZKyxsI23uu3IHLOkRHJBzkYFe5anfxaXpV5qEwZorWB53CkAlVUscbiB27kD3FeLfs9wT6jP4m8S36CW7vLlUN0YlGXJaSUAjpkshIAA4XrjgA9wA5HTgU+iigAoPQ0h6GvM/it8SrDwrol7pllqH/FQyqEijgwzW+QDvc9FO3kA88jAxyACn8V/i2PB+/RNHVJdaliBMpIZbTPQlecvjkKfUE5GAfnzwxpmseJdcXSdITfqV6zbrtpnVkjKsJdzA42MGO7IJOAB1IZugeH9b8deIzZ2Cm6vpyZp555OFBb55JGOT1bJPJOeASa+sfBvgHQvBFlHFp8Ae8MZjlvZY186QE7sZA4GSOPQDOcZoApfDv4baZ4CsAy7LrV5l23N9tIJGSQqgk7QOM4xuIBPYDuq+fPjx4jvNE8RafbaNq2r2N1Lbm4uTBfzLEykhECqH2gjy3JwozkHnJrrfgT4kvtf8G3Sapqcl9eWl6ygzyb5ViZVK7iTuOWMgBPoQOFwAD1U/dNfOetXEnwW+LhvrMSt4e1VTNLZwFVGwlsqFxjKMdy9PlO3Iy1c78SPiXrmoeNtRXRvEN5BpdvJ5VsLG6ZEIAAZspjeCwJBOeDwcYr1bU/hq+v8Awkhtbxr+68SraLdxz6jcNNKlwVDPEN7BUU/cxwB8rHJXNAHqkUiyxpJGyujAMrKcgg9x6ipT0ry74L+NP+Eg8NLouoz/APE70rMMkcpxK8IICuRgdMhD1OVBPLCvUT0oA8F8NGDw7+09rWnxrJINSSTa7EZVpES5Ynj7uVYD8Oepq146/wCTkvB3/XpF/wCjJqT4nNfaF8afB3iPelrp58m0e7dk2qDK4lBz0Hly9SOM8HIqbx1byp+0R4MuGicQvbpGkhU7WZZJSyg9yAykj3HrQB7XRRRQAHocda8f+Oviu5sdFtfC2lq0moa0djpCxMoi3AbQqndmRvkHBBAcc9vWb66jsbC5vJQxjgiaVwgy2FBJwO54rwn4WwXfxD+ImpeN9dtWmtLVjHYLMwdLeTcGRF6DKKeu37zbuG5oA9U8A+FF8G+ELHR90T3CAyXMsSgCSVjljnAJA+6CecKPpXU0UUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFB6UUUAYGveD/AA94oRhrWj2t25QJ5rJiUKG3BRIuGAzngHHJ9a8o1n9nCwcK2ia5cwkDDJfRiXec9dy7cD8DXu1FAHhFuvxv8HvcRRxw+IbC2jYo87rMWB+csPmWZ2HKgHPoAeKtQftE2ENy9trHhrULKaJ2jljSRZGjIyMEME5yMEHGOa9tPQ1lav4f0fX0RNX0uzvggZU+0Qq5QNjO0kZXOByPQUAUfDfjbw34r3roerQXckZO+H5kkAGMtscBiuWHzYxk9c8V0deUeI/gJ4U1eAnSll0a7yTviZpY2yQTujZugAIG1lxnvgCuOuPCnxj8Gy/ZtC1W41LTrdf3DxzI4VB8qr5UvIIUA7VBAzgE4oA+hz0Ncz4q8C+HvGVuY9Y06OScLsS7j+SaIYOMOOoBYkKcrnkg15bpvxt8Q+G54LHx54auYuNv2mKIwyuFGC2x8K5LAZKlVAPA6Cu00T40eCdZdom1NtPkWQqq3yeUHHGGDZKhSTwCQRgkgDmgDhJ/gh4o8LajLqvgnxGRLHlo4pCYZHQEsIyeUkztUYYKpPUAVYs/jdrnhi5XSPHfhudLyLANxABG8ij5d+w/K+SHO5WCnoABzXttjfWmp2cd3Y3UF1bPkJLBIJEbBIOGBIOCCPwNQ6voum67Z/Y9VsLe9t924Rzxh1DYI3DPQ4JGRyM0AZPhLx54f8aW3maReqZwC0lpKQs8YGMkpnkfMPmGVyetdPXkHib4G6dLJ/aXg+7m0PVYzviCTOYicnJB+8jcjkEgAYC965WPX/i98PL521ayuta0yJSZHdTPH5SHmQSp8yfKDgv2OSvHAB9E0V5t4f8AjZ4M1pUWe/fS7kttMN8u0dBzvGUAySOSDwcgDFd9Y31pqdpHd2N1BdWz5CSwSCRDgkHDAkHBBH4GgC3RRRQAUUUUAB6Vy3jfwVp/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']	Multimodal	Competition	False				Theorem proof	Geometry	Math	Chinese
73	"如图, 在锐角 $\triangle A B C$ 中, $D 、 E 、 F$ 分别为边 $B C 、 C A 、 A B$ 上的点, $L 、 M 、 N$ 分别为线段 $E F 、 F D 、 D E$ 的中点. 

<image_1>
证明：若 $A D 、 B E 、 C F$ 为 $\triangle A B C$ 的三条高线, $X 、 Y 、 Z$ 分别为线段 $E F 、 F D 、 D E$ 上的点,且满足 $D X / / A L, E Y / / B M, F Z / / C N$, 设 $\triangle A B C$ 、 $\triangle X Y Z$ 的外接圆半径分别为 $R 、 r$, 则

$$
r=2 R \cos A \cdot \cos B \cdot \cos C,
$$

其中, $A 、 B 、 C$ 表示 $\triangle A B C$ 的内角."	['先证明一个引理.\n\n引理 设 $\\triangle A B C$ 的内切圆 $\\odot I$ 与边 $B C$切于点 $D, \\angle A$ 内的旁切圆为 $\\odot I_{1}, M$ 为边 $B C$的中点, 则 $A D / / M I_{1}$.\n\n证明 如下图, 设 $\\odot I_{1}$ 与边 $B C$ 切于点 $E, E E_{1}$ 为 $\\odot I_{1}$ 的直径, 过点 $E_{1}$ 作 $B C$ 的平行线, 与直线 $A B 、 A C$ 分别交于点 $B_{1} 、 C_{1}$. 则 $B_{1} C_{1}$与 $\\odot I_{1}$ 切于点 $E_{1}$.\n\n<img_4173>\n\n因为 $A$ 是 $\\triangle A B C$ 与 $\\triangle A B_{1} C_{1}$ 的位似中心, $D 、 E_{1}$ 为对应点, 所以, $A 、 D 、 E_{1}$ 三点共线.\n\n由 $B D=C E$, 知 $M$ 为线段 $D E$ 的中点.\n\n于是, $D E_{1} / / M I_{1}$, 即 $A D / / M I_{1}$.\n\n引理得证.\n\n因为 $B 、 C 、 E 、 F$ 四点共圆, 所以,\n\n$\\angle A E F=\\angle B$.\n\n类似地, $\\angle C E D=\\angle B$.\n\n因此, $E A$ 为 $\\angle D E F$ 的外角平分线,且\n\n$\\angle D E F=180^{\\circ}-2 \\angle B$.\n\n类似地, $F A$ 为 $\\angle E F D$ 的外角平分线.\n\n从而, $A$ 为 $\\triangle D E F$ 中 $\\angle D$ 内的旁心.\n\n又 $D X / / A L, E Y / / B M, F Z / / C N$, 由引理知 $X$ 为 $\\triangle D E F$ 的内切圆 (圆心为 $\\triangle A B C$ 的垂心 $H)$ 与边 $E F$ 的切点.\n\n类似地, $Y 、 Z$ 为 $\\triangle D E F$ 的内切圆分别与边 $F D 、 D E$ 的切点.\n\n从而, $\\triangle X Y Z$ 的外接圆就是 $\\triangle D E F$ 的内切圆.\n\n设 $\\triangle D E F$ 的外接圆半径为 $R_{1}$.\n\n因为 $\\triangle D E F$ 的外接圆是 $\\triangle A B C$ 的九点圆, 所以, $R=2 R_{1}$.\n\n由于 $\\angle D E F=180^{\\circ}-2 \\angle B$, 类似得\n\n$\\angle F D E=180^{\\circ}-2 \\angle A$,\n\n$\\angle E F D=180^{\\circ}-2 \\angle C$.\n\n故 $r=4 R_{1} \\sin \\frac{\\angle F D E}{2} \\cdot \\sin \\frac{\\angle D E F}{2} \\cdot \\sin \\frac{\\angle E F D}{2}$\n\n$=2 R \\cos A \\cdot \\cos B \\cdot \\cos C$.']			['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b+EDsBQBkfDvw/ceF/AWkaPdsTcwRFpRgDY7s0jLwSDtL7cg84zxmuqoooAKKKKACiiigAooooAKKKKACiiigAooooAKD0NFNcBkYEAgggg96APlP4sa3pnif4swq9/ANIt0t7R762bzh5RO93+XOSPMcYH92vbvFfxb8K+HdHluLXV7PUb5o2+zW1pKJt7jAG8rwgyRnJBIBxkjFb7+CvCkkrSP4Y0VnbLFm0+Ikn1J21aXwt4eS6a5TQtMW4ddrSraIHI44zjPYflQB4z8CvBV1Pqlz4z123vPtJObGW4LqZTIpMkxzy+VYYbJB3N1IyPfaaAAeB3/pTqACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoPSiigDzXxL8FPCvifXJ9Wn+3WVxPhplsnREkfJy5DI3zHvg89cZJJ1PCfwt8K+DbmG806yeXUI1ZBeXMhkkwx544RTj5cqoOOM8nPbUUAFFFFAH/9k=']	Multimodal	Competition	False				Theorem proof	Geometry	Math	Chinese
74	"设椭圆 $C$ 的两焦点为 $F_{1}, F_{2}$, 两准线为 $l_{1}, l_{2}$, 过椭圆上的一点 $P$, 作平行于 $F_{1} F_{2}$ 的直线, 分别交 $l_{1}, l_{2}$ 于 $M_{1}, M_{2}$, 直线 $M_{1} F_{1}$ 与 $M_{2} F_{2}$ 交于点 $Q$.证明: $P, F_{1}, Q, F_{2}$ 四点共圆.

<image_1>"	['设陏圆方程为 $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1,(a>b>0)$, 据对称性知, 点 $Q$ 在 $Y$ 轴上 (如图); 记 $Q F_{1}=Q F_{2}=m$,\n\n$P F_{1}=r_{1}, P F_{2}=r_{2}, P Q=t, M_{1} F_{1}=M_{2} F_{2}=k$, 则有:\n\n$\\frac{P F_{1}}{P M_{1}}=e, r_{1}+r_{2}=2 a$, 为证 $P, F_{1}, Q, F_{2}$ 四点共圆, 据托勒密定理, 只要证, $m r_{1}+m r_{2}=t \\cdot F_{1} F_{2}$, 即 $m \\cdot 2 a=t \\cdot 2 c$, 也即 $\\frac{m}{t}=\\frac{c}{a}=e \\cdots(1)$\n\n由 $\\frac{Q F_{1}}{Q M_{1}}=\\frac{O F_{1}}{H M_{1}}$, 即 $\\frac{m}{m+k}=\\frac{g^{2}}{\\frac{a^{2}}{c}}=\\left(\\frac{c}{a}\\right)=e^{2}$, 所以 $\\frac{k}{m+k}=1-e^{2}$,\n\n在 $\\Delta P M_{1} Q$ 中, 由斯特瓦特定理,\n\n$P F_{1}^{2}=P M_{1}^{2} \\cdot \\frac{m}{m+k}+P Q^{2} \\cdot \\frac{k}{m+k}-m k \\cdots (2)$\n即 $r_{1}^{2}=\\left(\\frac{r_{1}}{e}\\right)^{2} \\cdot e^{2}+t^{2}\\left(1-e^{2}\\right)-m \\cdot \\frac{m\\left(1-e^{2}\\right)}{e^{2}} \\cdots (3)$\n\n因为 $1-e^{2} \\neq 0$, 由(3)得, $\\frac{m^{2}}{t^{2}}=e^{2}$, 即 $\\frac{m}{t}=e$, 故(1)成立, 因此 $P, F_{1}, Q, F_{2}$ 四点共圆.\n\n(也可不用托勒密定理证: 由(2)得 $P Q^{2}=m(m+k)$, 则 $\\triangle P Q F_{1} \\backsim \\Delta M_{1} Q P$, 于是\n\n$\\angle Q P F_{1}=\\angle M_{1}=\\angle M_{2}=\\angle Q F_{2} F_{1}$ , 因此 $P, F_{1}, Q, F_{2}$ 四点共圆.']			['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proof	Plane Geometry	Math	Chinese
75	"如图所示, $A D 、 A H$ 分别是 $\triangle A B C$ （其中 $A B>A C ）$ 的角平分线、高线, 点 $M$ 是 $A D$ 的中点, $\triangle M D H$ 的外接圆交 $C M$ 于点 $E$. 求证: $\angle A E B=90^{\circ}$.

<image_1>"	['连结 $H E$.\n\n由 $M$ 是 Rt $\\triangle A H D$ 斜边 $A D$ 的中点可知 $M A=M H=M D, \\angle M D H=\\angle M H D$.\n\n由 $M, D, H, E$ 四点共圆可得 $\\angle H E C=\\angle M D H=\\angle M H D$\n\n从而 $\\angle M H C=180^{\\circ}-\\angle M H D=180^{\\circ}-\\angle H E C=\\angle M E H$.\n\n又由 $\\angle C M H=\\angle H M E$ 可知 $\\triangle C M H \\sim \\triangle H M E$. 故 $\\frac{M H}{M C}=\\frac{M E}{M H}$, 从而 $\\frac{M A}{M C}=\\frac{M E}{M A}$\n\n又因为 $\\angle C M A=\\angle A M E$, 所以 $\\triangle C M A \\sim \\triangle A M E$. 故 $\\angle M C A=\\angle M A E$.\n\n由 $A D$ 是角平分线, 可得 $\\angle B A E=\\angle B A M+\\angle M A E=\\angle M A C+\\angle M C A=\\angle D M E$\n\n则有 $\\angle B H E+\\angle B A E=\\angle D H E+\\angle D M E=180^{\\circ}$, 从而 $A, B, H, E$ 四点共圆.\n\n所以 $\\angle A E B=\\angle A H B=90^{\\circ}$. 命题得证.']			['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAVECKgMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKbIypGXZgqqMkk4AFcJ4M+J1h4x8Taho1rYz2/2WLz4pZW/18eQAwXHAIZSPZhQB3tFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFI3THrS1Dd3MFlZzXVzIscEKNJI7dFUDJJ/AUAcf45uZtTlsfBtjIyXGr5N1InWGzU/vW/4Fwg/3jXHahBD4X/aJ0A2yLDZalpotdq9BtVlVR7DZHWz4d8MP4we48Y6nfatYXGpH/RI7O7a3MdmP9UrFeSTguf96uR+LGhJ4Ov/AAr4ki1DU7v7NqK7zfXbz7cEOAu48D5GoA97FLTUIYBlIKkZBFOoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAqhrWkWWvaRcaZqUJms7gBZYxIybgCDjKkHGQO/PSr9V768tdPspbu9uIra3iG55pXCqg9STwKAK+l6XbaRYx2Vp54touEWaeSUgem5yTj0GcegrM8Q+CNA8VlDrVrPdqmCsZvJkjBwRnYrhc8nnGa8t0r4uazrnxgtdLsnhPh+6mMcCSReWXQAnzAxAbJIJA6c46817oOpoAp6Xp1tpNjHZWgmEEY+QSzvMQPTc5Jx7Zq7RRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFICD0IoAWiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKQ02VlSMu7BUUZYnoBXnd74z1TxVeS6R4BSOVI3KXWuzrm2tz3Ef/PV/TGV6dQcgA3/FfjbTfC6Jbus15qtxxa6daLvnmPsOw9Sf16Vz9p4L1bxbdxat48dHiRxJbaDA+beH0Mp/5av/AOO9eoOBv+FPBGm+GDLdAyX2r3HN1qV2d00p+vZenyj0Gc4rR8Vat/YPhPVtVBAe0tJJUz0LhTtH4nA/GgD5X8RTXWs/FuS6s2+zSXN4V0t4sAExsYYCPQbolHsPpX1D4N8Sw+LPDFnqsYCSSJtuIh/yymXh1Ppg9M9Rg18/avoUmif2Z9nhIvNL8I22pBT1WUXyySZ+m969VsJo/CXxJ8uNiNC8WL9otz/DFegDcPbeuDnqWwO1AHpdFNX29KdQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQB4P8I/FmvwfEDU/Bmq6hNqVtbGdI5pmLOjRPtyCedpGeD04rqfiN8TZvDuoW/hzw9arfeI7sqoTqsW77uR3J7A4GDms/4O6F4esbDUPEttraaxqk4P2uZUKGEE7yu1vmBJGSx64GMYNcp8EIH8VfEfXfFl+N88AMi552STMRwfZVYD2NAHfaT8Ln1KAXfjvV77W7+X5ntvtDpbQnuFVcZ/Qe1X7v4PeB7iMrFo32STostrPIjL79SD+INd4DzS0AcF8LNG1HQ9M1q0v59QlSPVZ4rQ3zOWNumFRhu7N8xyOD2rvaKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKa+QvGc0AK3Tv17VieJPFWkeFNPF3q1yIwx2xQoN0kzf3UUck1z+tePJZdUfw94OtF1jWkOJpGb/AEazHrM46nIPyg5OCOuKn8M+AYtM1Aa7r12+t+I3HzXs4G2H/ZiTogGTz164xkigDIXRPEXxEkFx4nWXSPDmd0WixMRNcjqDcMOQOnyD9CM16FYWNrptpFZ2VvFb2sS7Y4olCqo9gKsiloAQ1wHxWzqGjaT4bQFjreqQWsgU4IhB3u34bRXfnpXA3qnWPjbpkBBMOhaXJdFgekszeWFP/AFJoAy/EGmrq/xe1PTM7RdeC5YAQOm64K8fnS6Zpf8AwsD4F6RHDIVv4LVGs584aO5gygOe2SpGfRia0h/ycHnr/wAUr/7d074aE2aeJtDkwP7O1q4ESDtDIRIn57moA2/A/iQeKfC1rqLr5d2Mw3cOMGKdOHXHbnkA84Iro683wPBXxUDA7NG8Unn+7DfL+g8wZ+rfSvR1oAWiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAPmj4L2v234jeJINOJXSpbS4jyv3ApkAj/HHT2BrV/Z/m/sPxR4l8OX48m/YR4jY45iZwwH/fYr2Lwp4O0bwZprWWj25jVyGlmkO6SYjoWPtzgDAGeBWP4t+GemeJdTi1m3ubnStbhOY7+zOGyOm4d8fUfXtQB2w61j+LPEMHhXwxfa1cR+atqm5Yt2DIxOFXPuSB0Nc9FoXxFgQQjxjpk4Ax582lYc++A+Kpy/DC91u+t7rxX4u1HVRbyLLHbQxJbQbgcjKDIP16+9AHoUDSPEjSxiOQqCyhshT3GcDP1qWmqMGnUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRSHp1xXH+JvHlto14mj6XbPrHiCYfutPtiPkz/ABSt0jXkZLevpkgA6LWNZ07QdNl1DVLuK1tYhlpJGwD7D1PoBya4DzfEvxM4t/tPh3wpIBmYjbeXyH+72jQ+vUjHUHAvaT4Cu9R1OHxB43uo9U1OM7reyQH7JZ/7in7zf7Te3oDXfKOevagDO0LQtM8OaZHp2k2cdrbRjG1Byx9WPUk+prToooAKKKKAENcB8Ox/aev+MvEjA/6Zqf2SI9mit12Kw+pLflXVeJ9WGheFtV1XjNpayTKD3YKSo/E4H41lfDfSDonw80SzcMJTbCaUMed8hMjZ/FiKAMn/AJuEH/Yqf+3dO0grpnxs8Q2eT/xNtMt9QAPYxsYSP5U3/m4Qf9ir/wC3dP8AFW7Tvif4L1RQFiuftOnTuep3JujH/fSmgDc8beG18V+FrrTQ/lXXEtpN0MU68owPbng47E1F4D8SN4n8MQXdyhi1GBmtr+E8GO4ThwR27HHYH610p6GvOrvd4K+KEV6oC6L4nK203ZYr1Qdh9t4yOOpyT0oA9Hopq06gAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKQ0ALUF5d29jaSXV3cRW9vEN0ksrhVQDuSeAKwfFXjXSvCkUSXLS3OoXB22un2y7552PQKo6D3P88CuctvCGseNbmPUvHREVmj77bQLeQ+VH6GZh/rG/T8yKAI5fEevePpHsvB4fTtDBKT6/OhDSDoVt0PX/AHjjHOMEA11vhXwjpPhKyeDToWMsp3XF1Md81w/dnbqTnJx0GT61tQQxQRJFDGscaKFREUBVA4AAFS0AFFFFABRRRQAUGikNAHBfFc/bfD+m+HlyTreqW9m+37wjD73b8An613aAKAAMADAA7VwWpf8AE4+NmjWgBMOiabNesR08yU+WFPvtGa7+gDgP+bhP+5U/9uqd8XFW38JWmtlSW0XU7W/AHfbIFI/JzTR/ycIP+xU/9u66Pxpph1jwTrenqm957KVY1/29pK/qBQBtoQwDDoRmsTxj4ci8VeFr7SJGCPKm6CXvFKvKNn2YDPqMjvUXgPUf7W8BaFel97yWMQkb1dV2t/48DXQP92gDlvh/4kl8S+GkkvVMeq2btZ6jEwwUnTAbI7Z6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proof	Plane Geometry	Math	Chinese
76	"如图, 在凸五边形 $A B C D E$ 中, 已知 $\angle A B C=\angle C D E=\angle D E A=90^{\circ}, F$ 是边 $C D$ 的中点,线段 $A D, E F$ 相交于点 $G$, 线段 $A C, B G$ 相交于点 $M$. 若 $A C=A D, A B=D E$,求证: $B M=M G$.

<image_1>"	['连接 $A F$, $B F$ 如下图.\n\n<img_4058>\n\n因为 $A C=A D, F$ 是边 $C D$ 的中点,\n\n则 $A F \\perp C D$.\n\n又 $\\angle C D E=\\angle D E A=90^{\\circ}$,\n\n故四边形 $A E D F$ 是矩形.\n\n所以 $\\triangle A C F \\cong \\triangle A D F \\cong \\triangle E F D$.\n\n因此 $\\angle E F D=\\angle A C F$,\n\n从而 $E F / / A C$.\n\n又因为 $A B=D E$,\n\n所以 $\\triangle A C B \\cong \\triangle E F D$, 因此 $\\triangle A C B \\cong \\triangle A C F$,\n\n故 $A B=A F, C B=C F$.\n\n连 $B F, B F$ 交 $A C$ 于 $N$,\n\n$A C$ 垂直平分线段 $B F, B N=N F$.\n\n线段 $A C, B G$ 相交于点 $M$, 因为 $E F / / A C$,\n\n由平行截割定理, $B M=M G$.']			['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P35DgIv4kjPtn0oA8j+IEr/Ef4v6Z4MtmdtN007r1k7HG6Q59l2oCejMR3r3iKNIoUjjVURAFVVGAAOwHavIPgN4bmh0a98W6hmS/1eRvLkblvKDHcfbc+T/wFTXsQ4FABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABTJEDqysoZWGCDT6KAPA/BzH4Y/Gm/wDCsxKaTrBBs8ngE5MXX33R+5xXvg6V5F8evDEl/wCG7fxHYgrfaPIHZ0+95RIyeOflba3sNxruvA3iePxd4O0/WFKiWSPbcIv8Eq8MPpnkexFAHR0UUUAFcv8AEHxSvg/wbqGq7gLhU8q1U/xStkL9cfePsprqK8F+Jc0nxC+K2j+B7NybKybzL1kPRiNzn0yqAAf7TEUAdH8CvC0mk+EX127Vjf6w/nbn+95QztyT6ks/uGX0r1YdKZbwRW1tFBCipFEgREUYCqBgAVJQAUUUUAFFFFABXFfFz/klXiD/AK4L/wChrXa1xXxc/wCSVeIP+uC/+hrQBV+Cf/JItD/7eP8A0okrv64D4J/8ki0P/t4/9KJK7+gAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoozSZGQM9aADIFebfEn4q2ng/wD4lelqt94glAEcCjcsJOMF8c5OeFHJ46Dri/EH4rzvqP8AwivgdDfaxO3lSXMI3CI4+6nYsOcseFwe/wB3V+G/wmt/C7DW9bcah4hmy7Su29YGPJ2k/ec5OWP4e4BjeA/hTe6hqI8W+PXe91OY+ZHZz/ME9DIOmR2ToP0Hs4HHNA4ApaACiiigAooooAKKKKACiiigAooooAMivCfjBfT+MfHeifD/AE2Q4Eqy3ZHO1iuQf+AR7n993tXsev6zbeHtCvtXvDiC0iMrAEAtgcKM9ycAfWvJPgbo91q1/rPj3Vfmu7+Z4oGPoTmQj0Gdqj02kdKAPZrCxt9M062sLSMR29tEsUSDsqjAH5CrFIOlLQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAQXdrDe2s1rcRiSGeNo5EPRlIwQfwNeGfCy5m8CfEnWfAN9I32e4lMtkz/wATAZU/8CjAz7qBXvVeK/HfQri0Ok+ONLGy90yZEldeu3cDGx+jce+/2oA9pAwMUtZHhnXrbxJ4Z0/WbYjy7uFZNoOdjdGX6hgR+Fa2RQBi+LfEEHhbwxqGs3AylrFuVf7zn5UX8WIH4mvNfgRoE8llqfjLUv3l9q8ziORgMlNxLsP958/98CqHxm1G48VeLtE+H2lP88kqTXTD+FiDjP8Aupucj0K4r2nTdPt9K0y10+0j2W1rEsUS+iqMCgC2OlFFFABRRRQAUUUUAFcV8XP+SVeIP+uC/wDoa12tcV8XP+SVeIP+uC/+hrQBV+Cf/JItD/7eP/SiSu/rgPgn/wAki0P/ALeP/SiSu/oAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigApMjOKCwHU1Q1jWNP0LTJ9T1K5jtrWBcvI7Yx6ADuT2HegC3PPFbxPNNIkccal3d2CqqjqST2ArwjxZ8Qda+I2rN4R8BRyG2fIur/ldyd/m/gj9+pPA9DU1DV/E3xw1h9J0ZZNN8LW8oM875+cA5Bf+83GVQcA4J6Aj2jwn4Q0nwbpCadpFuETgyzNzJK/95j3+nQdqAMn4f8Aw30rwHYH7OBc6jKuJ711wzf7Kj+FenH4knjHaDpQoIUA9qWgAooooAKKKKACiiigAorP1fW7DQbNrvUpXhtl5aUQu6oPVioOB7mpNL1Wy1rTYNR064W4tJwTHKoIDAEjv7g0AXKKgvb2206ymvLyZIbeFC8kjnAVR1NUdF8SaV4it/tGlXD3MBGRMIJFRuccMygHn0oA1aM0daq399b6bY3N7dSCO3t42lkY/wAKqMk/oaAPHPjlrN1q2oaN4C0lt1zqEySXCj0JxGp9shmPoFBr1rQNEtvD+gWOkWg/c2kKxK2MFvVj7k5J9zXjnwhsbjxj471v4g6jGcCVorMMc7WYYwP92PC/8CNe7DpQACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKoaxpltrWkXml3a7re7haGTHUBhjI9xnIPar9NI5oA8P+CeqXXh7xDrfw/1V8T20zS23oSOHAz2YbXH/AAI969i1nVbbQ9GvNUu2229rC0rkdSAOg9z0+teN/GjTbjwt4r0T4g6UmJYpUhugDjcR93djnDKGQn0CjvSfGLxb/wAJNpnh7wx4fl82TXDFdMAcfu2IEat6Zbk+mwUASfBHSrrX9b1r4gasoa4u5nits9BnlyM9gNqAjsGFe4DgAGsrw9odt4a8OWGj2gzDaRCMHHLnqzfUnJP1rkLrxVqfizxZdeGvC1wtra2GP7T1fYHKH/nnCp4LcH5j0weOBkA9ELAdelGRmuZ/4Qyx8jnU9dFyR/x8nVp9+eOdu7Z+G3b7U3wPZaxZ6TdjW7+e+uWvpwss3yny0by0+UAAAhN3vuz3oA6mikHSloAKKKKACuK+Ln/JKvEH/XBf/Q1rta4r4uf8kq8Qf9cF/wDQ1oAq/BP/AJJFof8A28f+lEld/XAfBP8A5JFof/bx/wClEld/QAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFJkDOT0pc1xPxA+JGleBLEGYi51KVc29ijYZu25j/AArnPPfBAB5wAa3i3xdpPg7SzqOrXBRSdscKDMkzYztUd/0HqRXi+naR4m+OOsLq2svJpvheCT9xAnR+eVTgbm45cjAPQcYF3wp8PNa+Iurr4v8AHrP9lkG61sOVDpnKjGcpH7dW655yfdoYI7eFIYY0jijUKiIoUKo6AAdBQBU0jR7DQtMg07S7WK2tIBiONBwPfPUk9yeSTmr4GBilHSigAooooAKKKKACiiigAooooA5P4gRvf6BHoULMs2sTrZZXqEwXk/8AHEcfiKwfgXd/aPhZYxZybaeeH8d5f/2aujcjUfiIo6xaRY5Pp51w2B+ISI/9/K8z+G17eQweIvBmls8V9/bVxuuAMi0txtVn9N3ykKO5bPRTTAl+NWtXWsaI2m6c7/Ymvo7FfLPN3c5JKD1RNuD6uw/uV67oekxaJoVhpcAXy7SBIgQPvFQBn6nr+NeYapp9tqPxk8K+GbGJV03w5aNevGv8LHG36nIjOTydxr2AdKAEyB1rx749eJJYdIsvCenlpL7V5F3xx8t5YYAKB6s+AP8AdYV69LJHCjyyuqRoCzO5wFA6knsK8G+H8cnxH+Lup+MrpHOnaadtmr54PKxjHqFy5/2iD3pAev8Ag3w5F4T8J6fo0O1jbxDzJF6PIeXb6Ek49sVvUg6UtABRRRQAUUUUAFFGaKACiikyKAFooyMZrIv/ABX4d0skX+u6bbMP4ZbpFP5E5oA16K4G++M3gOxcoddWdxztt4JJM/iBj9a5i9/aM8MxblstM1S4cHgskcat+O4n9KAPZc0Zrwv/AIXT4w1WPOg/D65fP3ZCss6/+Oov86VtU+O2tFfI0q00tG6uqxLt+okZm/TvQB7nkGo5Z4oELzSJGg6s5wB+JrxE/Dn4taxKG1bxytqncWtxID/3yiov61JD+ztDcTmbWvFV9eyf3o4gjfm7P70AelXvxB8H6eG+0eJtLBXqqXKuw/BSTXMX3x38CWisYb+5vCP4YLVx/wCh7RUdj8BPA9oB59te3xHe4uiD/wCOba6ay+HHgvTlAt/DWmnb0aaATEfi+TQB5F41+OGheJ/Dd/olv4fv50uYiA07pGY2HIcBd33SM/h7msf9n3TLLUPG9xeXcqvc6fa77WFs5+Y7S49lBxj/AGwe1d78b9dTR/Ddn4V0aBEu9Yk2eTbKAfKBHyhVx99iF9wGFcTrfhmX4M+IfCniK13y25jWLUAGyHlx+8A7YZWO0eqHPSgD3/xTfTaZ4S1m/t8+dbWM8yY7MqEj9RXC/Aezjt/hxHecGa9uZpZnPUkNs5P/AAHP4mvSAbXU9PyGjuLS5i6jlZEYfqCD+teceFdI1/4aNeaSNMuNY8PyTNNZz2joZoM8FHRiufXK55z68MDZ+IvjHUPBOjJqtva2d3C0yQCGR2VyzZ6YyCODXT6M+pz6dFLq0NvDcuoYxQMWCZH3SSBkiuC8Q6TrPj/xB4eRtHudP0LTbsXV19vZFacrjaqorMem4c4+9Xpw6UgCiiigAooooAK4r4uf8kq8Qf8AXBf/AENa7WuK+Ln/ACSrxB/1wX/0NaAKvwT/AOSRaH/28f8ApRJXf1wHwT/5JFof/bx/6USV39ABRRRQAUUUUAFFFFABRRRQAUUUUAFFFJkZxnmgBaQsAQCeTRuGcZ5614v49+K13qOpDwn4ARr3U5yY5Ly35CeojPTI5y/Rex7gA2fiR8WLfwvIdF0RFv8AxBLhFjVd6wE8DcB1fnhRz0zwRnM+H/wouXv/APhK/HDtfaxM3mpbT4YREj7z9i3TC9Fx3OMbPw2+FFp4QVdV1J0v9ekGWuD8yw5zkR57nOCx5PIGATn0lRgY9KAAcCloooAKKKKACiiigAooooAKKKKACopo2kjdVkeNmUgOuMqSMZGcj8wRUtGaAOd0bwqdF1S/v11nU7yS+cSTrdGEqzBQoxtjUgAAAAGptN8L2GjnVZNPDw3Op3D3NxcDaXLt6ZBGB2BBHX1rcozQBxek/Dm20jxXd+JItc1iXUbwbbhpmgZZVyDgjyhgfKPu4wBxXZjgAUuQe9RTTwwIXmkSNB1Z2AH60AeZfHTxUdE8F/2Vavi+1djAAv3hEP8AWH8chf8AgRrpPhx4VHg/wTYaayAXTDz7sjvK3JH/AAHhfoteMyeJtF8XfG59Z1rU7e30HR+LXzm4lMZ+XA6nL5fp0AB7V6Re/HfwJaITDf3N4w/hgtXB/NwtAHpYPbvS5rxK6/aM055Fi0jw1qF5Kx4SeRYj+Sh81Efid8UNWk26N4CaBW6Nc28rf+PHYtAHuORS5rw1bT476y58y6tNJjP8JMAH/jodhSL8HfHurIRrvxBnAb70cUs0y/gCyD9KAPZ73VtN01C99qFraqOrTzKgH5mua1D4q+BtNGZvEtk//XsTP/6AGriLP9nDw+gzf61qdw/UmLZGD+YY/rXTWXwS8B2ZRjozXDr/ABT3MjZ+oDAfpQBk3/7QXgy0YrbpqV6exhgCr/4+yn9KxR8f73U5THoHgm8vSOh8wuR/wFEP869WsfBvhrTGD2Xh/S7dx0eO1QN+eM1tKgUYAAHoKAPDV8ZfGjW0P9n+EoLNScq0sBjcfXznx+lL/wAIr8btaj/03xPbWAbqqzCNh/35T+te5YNLQB4a3wF1nVSh8QeOru7APzIUeX8meT+la9j+zx4Qt3DXNzql2R/DJMqqf++VB/WvW6KAOJsfhJ4G05t0Phy1kP8A08M836OSK6aw0PSdLTZp2l2Vmn923t0jH/joFaFFADdv4UuKWigAooooAKjkdUVnYgKoySTjA9akJxXmXxw8VHw94Gksbdyt7qpNtHtPIj48w/l8v/AqAOQ8FI/xN+M2oeK5xv0rSCEs8g4yMiLGf+BSH0YivU/H3hZPGHgu/wBJ2r9oZPMtmI+7KvK/TPIPsxqt8MPCf/CIeBrKxkj23kw+0XeevmNj5f8AgIwv4V2JBIoA8m+BHiiTVfCk2hXbEX+kSeVsfIYwnO3j/ZO5fYBa9ZHI46eteBeJ1/4Vf8cLPxBHmPR9byLkDhQWIEn5Nsk/HFe+gjaCDkHpigAx/k0tFFABRRRQAUUUUAFcV8XP+SVeIP8Argv/AKGtdrXFfFz/AJJV4g/64L/6GtAFX4J/8ki0P/t4/wDSiSu/rgPgn/ySLQ/+3j/0okrv6ACiiigAooooAKTI9elLXnHxk1G60nwrBPpd5eW2rXN3Fa2ht7hk3MTkgqDg8Ajkd6APRsilBz0ry3xxa6r4H8ENr2neJdSa+sRF5sd3N58VzllVgVbkfeJ4I6fiPQfD+pSax4c0zU5YfJku7WOdo/7hZQxH60AaNFFGaADIqC5uoLS3kuLiaOKCNSzySMFVAOpJPQDBqnrmuad4e0ufUtUuEgtIRlnc9f8AZA6knsB1NeEXV/4n+OWtNZaesumeEreQCR26Pg5G7H337hAdq8Z9aALfiXx3rvxR1aTwp4FR49NIxdXzZQumeSx6pH7febp0yD6d4E+Hek+BdMMNov2i9lUfaL2RcPIfQf3VyBxn6knmtXwv4V0vwjo8em6Tb+VEvLsTl5W/vMe5P5DoABW2OABxn2oAB0paMijNABRSZFLmgAoqne6tpumoXvtQtbVR1aeZUA/M1zWofFXwNpozN4lsn/69iZ//AEANQB2OaK8nv/2gvBloxW3TUr09jDAFX/x9lP6Vij4/3upymPQPBN5ekdD5hcj/AICiH+dAHuVJkV4YvjL40a2h/s/wlBZqTlWlgMbj6+c+P0pf+EV+N2tR/wCm+J7awDdVWYRsP+/Kf1oA9zyMZrIv/Ffh3SyRf67ptsw/hlukU/kTmvJG+Aus6qUPiDx1d3YB+ZCjy/kzyf0rXsf2ePCFu4a5udUuyP4ZJlVT/wB8qD+tAG7ffGbwHYuUOurO45228EkmfxAx+tcxe/tG+GYg62Wl6pcuD8u9UjVvx3E/pXX2Pwk8Dac26Hw5ayH/AKeGeb9HJFdNYaHpOlps07S7KzT+7b26Rj/x0CgDxv8A4XV4x1aPOhfD65fP3ZCss6/+Oov86VtT+O2t7Ps+mWmlIx5cLCuB7iRmb9K9x2/hS4oA8OPw5+LOsSb9W8ci1Q9VtriQf+OoqL+tcd8Rvhjb+CvD39qap4nuNS1OeVYraNotu49WJJZjgDPTHJHrX1EfxrwHVyfin8dINMGZND0EEy91YqRv9vmfah9VXIoA1fAfwP8ADs3hPT73xHYzXGo3MQmeMzvGIw3KrhSDkDGc9816DY/DfwZpyqLfwzphK9GmgEpH4vk11A4FLQBBBaW9rGI7aCKCMfwxoFH6VNilooAB0ooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAGk4OK8CtCPin8eWusrLoWgAbCDlX2N8vI4O6TJ91WvRfi14tPhHwLd3EEmy+uv9GtSDgq7A5b/gKgke4HrVT4MeEz4Z8B28s8ZW+1Ii6m3DBUEfIv0C84PQsaAPRBwBS0DpRQBwvxY8Kf8JZ4FvbeGPdfWo+02uByXUElf+BLlcepHpVD4L+LR4l8CQW9xKrX2mYtZhn5ioH7tvxUYz3KmvRyCTXgcIb4W/HdoeYtD1/7nOEXe3Ht8kmR7K2e9AHvo6UtIDxS0AFFFFABRRRQAVxXxc/5JV4g/wCuC/8Aoa12tcV8XP8AklXiD/rgv/oa0AVfgn/ySLQ/+3j/ANKJK7+uA+Cf/JItD/7eP/SiSu/oAKKKKACiiigBMgV5N40iuPFXxh8P6DZ3IgGj27anNLsDhHyNmVPBwVT/AL6r1S5kkhgeSO3knYdI4ioZvpuIH5mvLPDFn4r0zxl4l8R6n4PvpbjU3VbYRXlqTFCoICnMo5wE6Z6UAZXi2O70/wAe+H9O8cXja5oWozAWccai3SKYELmSNfvgb15J6MeO1e2RqqRqqABQMAAYA/CvM28IeIPGXjfTNf8AFFtb6bpuktvs9NjlE8jyZB3SMPlHKqcDP3QPevTVBCgHrQAEgdTXOeMfGukeCtKN/qc3LZEMCHMkzeij+ZPAroJdwRin3wDjjPOP1/MV8erb+LPiX46nVmFxqsbMzR3bLEIkQ/c8tzjAPVQD3JzyaAPQNJ8P+JfjVq6a54kkksPDULf6NboSA47hPy+aQ9TwB2HutlY6boGmRWdnFb2VlAuERcIqjv8AifXvzXjafDf4raltXUPGyWUIXaI7S4kXaMdNiKi4xU8P7O8FzcfaNY8VX16553JEFb/vpmf3oA9Ju/iD4P09D9o8TaWCvVY7lXYf8BUk1zN98ePAlorGG/ubwj+GC1cfq+0VHYfATwPaACe2vb3He4uiD/44FrprH4b+DNOUC38M6YSOjTQCUj8XyaAPO7r9ozTnkWLSPDWoXkrHhJ5FiP5KHzUR+J3xQ1aTbo3gJoFbo1zbyt/48di17XBaW9rGI7aCKCMfwxoFH6VNigDw5bT476y58y6tNJjP8JMAH/jodhSL8HfHurIRrvxBnAb70cUs0y/gCyD9K9zHSigDxez/AGcPD6DN/rWp3D9SYtkYP5hj+tdNZfBLwHZlGOjNcOv8U9zI2fqAwH6V6FRQBh2Pg3w1pjB7Lw/pdu46PHaoG/PGa2lQKMAAD0FOooATBpaKKACiiigAooooAKKKTIoA5L4leKl8IeCL/UVcC7dfItBnkyvkAj6DLf8AATXNfAvwo2ieC/7Wuk/07WGFwWbr5P8AAD9clv8AgQrl/iBK/wAR/i/pngy2Z203TTuvWTscbpDn2XagJ6MxHeveIo0ihSONVREAVVUYAA7AdqAHjpzRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABSEgHtS1y/xB8Ur4P8G6hqu4C4VPKtVP8AFK2Qv1x94+ymgDyrxIT8T/jja6CmJNG0Mk3AB4O0gy5+rbI/wyO9e+KAFAAwBwK8p+BXhaTSfCL67dqxv9Yfztz/AHvKGduSfUln9wy+lerDpQAtFFFABXmfxv8ACn/CQ+BpL6CMG+0om5jIHzGPH7xfy+b6oK9MpkiB1ZWUMrDBBoA474XeLP8AhLvA1neSyBr23/0a7HfzF4z+I2t+PtXaCvA/BzH4Y/Gm/wDCsxKaTrBBs8ngE5MXX33R+5xXvg6UAFFFFABRRRQAVxXxc/5JV4g/64L/AOhrXa1xXxc/5JV4g/64L/6GtAFX4J/8ki0P/t4/9KJK7+uA+Cf/ACSLQ/8At4/9KJK7+gAooooAKKKKACiiigAooooAawyCPWvHfix8P7sXK+N/Cm+31qyxNcRwjmUD/loo7sB1H8Q9+D7JTWB7DNAHFfDf4gWnjvQRNlItStwFu7cHG09mUf3Tz9Onau2HSvBviF4T1D4d+JE8feEhst9+b60VfkXJ5+Uf8s26Efwkgjtj1vwf4t07xl4dg1bT2wrfLLCTloZB1Rv5+4IPegDfooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACsDxl4jh8JeFdQ1qbaxt4/3UbH78hwEX8SRn2z6Vv5FeE/GC+n8Y+O9E+H+myHAlWW7I52sVyD/wAAj3P77vagDU+A3huaHRr3xbqGZL/V5G8uRuW8oMdx9tz5P/AVNexDgVXsLG30zTrawtIxHb20SxRIOyqMAfkKsUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFeC/EuaT4hfFbR/A9m5NlZN5l6yHoxG5z6ZVAAP8AaYivX/FviCDwt4Y1DWbgZS1i3Kv95z8qL+LED8TXmvwI0CeSy1PxlqX7y+1eZxHIwGSm4l2H+8+f++BQB7DbwRW1tFBCipFEgREUYCqBgAVJQOlFABRRRQAUUUUAeRfHrwxJf+G7fxHYgrfaPIHZ0+95RIyeOflba3sNxruvA3iePxd4O0/WFKiWSPbcIv8ABKvDD6Z5HsRW5d2sN7azWtxGJIZ42jkQ9GUjBB/A14Z8LLmbwJ8SdZ8A30jfZ7iUy2TP/EwGVP8AwKMDPuoFAHvVFIBgYpaACiiigArivi5/ySrxB/1wX/0Na7WuK+Ln/JKvEH/XBf8A0NaAKvwT/wCSRaH/ANvH/pRJXf1wHwT/AOSRaH/28f8ApRJXf0AFFFFABRRRQAUUUUAFFFFABRRRQBFNBHcwyQTxrJDIpR43AKup4IIPUY7V8/axp2pfA7xomt6Sktx4Xv3CTQZJ2DrsJP8AEOqsevIPfP0NVDV9Istc0u603UbdZ7S5QpJG3GffPUEcEE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proof	Plane Geometry	Math	Chinese
77	"如图, $P_{k}(k=1,2,3, \ldots, 100)$ 是边长为 1 的正方形 $A B C D$ 内部的点. $E, F, G, H$ 分别是 $A B, B C, C D, D A$ 的中点, 记 $d_{1}=\sum_{k=1}^{100} E P_{k}$, $d_{2}=\sum_{k=1}^{100} F P_{k}, d_{3}=\sum_{k=1}^{100} G P_{k}, d_{4}=\sum_{k=1}^{100} H P_{k}$.

证明：$d_{1}, d_{2}, d_{3}, d_{4}$ 中至少有两个小于 81.

<image_1>"	['<img_4229>\n\n如上图建立坐标系, 以点 $F, H$ 为焦点作经过点 $A$ 的椭圆.\n\n由对称性, 正方形 $A B C D$ 为椭圆的内接正方形. $P_{k}(k=1,2,3, \\ldots, 100)$ 在正方形 $A B C D$内部，则也在椭圆内部\n\n椭圆长轴长 $2 a=A H+A F=\\frac{1+\\sqrt{5}}{2}<1.62$,\n\n延长 $H P_{k}$ 交椭圆于点 $Q_{k}$, 连 $F Q_{k}$,\n\n则 $H P_{k}+F P_{k}<H P_{k}+P_{k} Q_{k}+F Q_{k}=H Q_{k}+F Q_{k}<1.62$,\n\n其中 $k=1,2,3, \\ldots, 100$\n\n所以 $d_{2}+d_{4}=\\sum_{k=1}^{100} H P_{k}+\\sum_{k=1}^{100} F P_{k}=\\sum_{k=1}^{100}\\left(H P_{k}+F P_{k}\\right)<162$,\n\n所以 $\\min \\left\\{d_{2}, d_{4}\\right\\}<81$. 同理 $\\min \\left\\{d_{1}, d_{3}\\right\\}<81$.\n\n所以 $d_{1}, d_{2}, d_{3}, d_{4}$ 中至少有两个小于 81.']			['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proof	Plane Geometry	Math	Chinese
78	"已知 $F$ 为椭圆 $C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ 的右焦点, 点 $P$ 为直线 $x=4$ 上动点, 过点 $P$ 作椭圆 $C$的切线 $P A 、 P B, A 、 B$ 为切点.

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求证: $A 、 F 、 B$ 三点共线;"	['$F(1,0)$, 设 $P(4, t), A\\left(x_{1}, y_{1}\\right), B\\left(x_{2}, y_{2}\\right)$ .\n\n则切线 $P A, P B$ 的方程分别为 $\\frac{x_{1} x}{4}+\\frac{y_{1} y}{3}=1, \\frac{x_{2} x}{4}+\\frac{y_{2} y}{3}=1$ .\n\n由切线 $P A, P B$ 过点 $P(4, t)$, 得 $x_{1}+\\frac{y_{1} t}{3}=1, x_{2}+\\frac{y_{2} t}{3}=1$, 即 $x_{1}+\\frac{t}{3} y_{1}=1, x_{2}+\\frac{t}{3} y_{2}=1$ .\n\n由此可得直线 $A B$ 方程为 $x+\\frac{t}{3} y=1$ .\n\n易知直线 $A B$ 过点 $F(1,0)$ .\n\n$\\therefore A 、 F 、 B$ 三点共线.']			['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proof	Conic Sections	Math	Chinese
79	"如图, $O 、 H$ 分别为锐角 $\triangle A B C$ 的外心、垂心, $A D \perp B C$ 于 $D, G$ 为 $A H$ 的中点.点 $K$ 在线段 $G H$ 上, 且满足 $G K=H D$, 连 $K O$ 并延长交 $A B$ 于点 $E$ .

<image_1>
证明:  $E K / / B C$;"	['<img_4189>\n\n如上图, 连 $B O$ 并延长交圆 $O$ 于点 $F$, 由 $O$ 为 $\\triangle A B C$外心, 知 $B F$ 为圆 $O$ 的直径.\n\n$\\therefore A F \\perp A B, F C \\perp B C$ .\n\n结合 $H$ 为 $\\triangle A B C$ 的垂心, 得 $H C \\perp A B$ .\n\n$\\therefore A F / / H C$ .\n\n同理, $F C / / A H$ .\n\n$\\therefore$ 四边形 $A H C F$ 为平行四边形, $F C=A H$ .\n\n作 $O M \\perp B C$ 交 $B C$ 于点 $M$, 则 $O M=\\frac{1}{2} F C$ .\n\n因此, 由 $G$ 为 $A H$ 的中点, $G K=H D$, 可得\n\n$K D=K H+H D=K H+G K=G H=\\frac{1}{2} A H=\\frac{1}{2} F C=O M$ .\n\n结合 $K D / / O M$ ，得四边形 $O M D K$ 为平行四边形.\n\n$\\therefore O K / / M D$, 即 $E K / / B C$ 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']	Multimodal	Competition	False				Theorem proof	Plane Geometry	Math	Chinese
80	"如图, $O 、 H$ 分别为锐角 $\triangle A B C$ 的外心、垂心, $A D \perp B C$ 于 $D, G$ 为 $A H$ 的中点.点 $K$ 在线段 $G H$ 上, 且满足 $G K=H D$, 连 $K O$ 并延长交 $A B$ 于点 $E$ .

<image_1>
证明:$G E \perp G C$ ."	['连 $B O$ 并延长交圆 $O$ 于点 $F$, 作 $G N \\perp A B$ 于 $N$, 如下图.\n\n\n<img_4071>\n\n由 $O$ 为 $\\triangle A B C$外心, 知 $B F$ 为圆 $O$ 的直径.\n\n$\\therefore A F \\perp A B, F C \\perp B C$ .\n\n结合 $H$ 为 $\\triangle A B C$ 的垂心, 得 $H C \\perp A B$ .\n\n$\\therefore A F / / H C$ .\n\n同理, $F C / / A H$ .\n\n$\\therefore$ 四边形 $A H C F$ 为平行四边形, $F C=A H$ .\n\n作 $O M \\perp B C$ 交 $B C$ 于点 $M$, 则 $O M=\\frac{1}{2} F C$ .\n\n因此, 由 $G$ 为 $A H$ 的中点, $G K=H D$, 可得\n\n$K D=K H+H D=K H+G K=G H=\\frac{1}{2} A H=\\frac{1}{2} F C=O M$ .\n\n结合 $K D / / O M$ ，得四边形 $O M D K$ 为平行四边形.\n\n$\\therefore O K / / M D$, 即 $E K / / B C$ .\n\n由 $H$ 为 $\\triangle A B C$ 的垂心, 知 $\\angle N A G=90^{\\circ}-\\angle A B C=\\angle D C H$, 结合\n\n$H D \\perp B C$, 得 $\\triangle A N G \\backsim \\triangle C D H$ .\n\n$\\therefore \\quad \\frac{N G}{D H}=\\frac{A G}{C H}, \\angle N G A=\\angle D H C$ .\n\n又 $G K=H D, A G=G H$, 因此, $\\frac{N G}{G K}=\\frac{G H}{H C}$ .\n\n又 $\\angle N G K=180^{\\circ}-\\angle N G A=180^{\\circ}-\\angle D H C=\\angle G H C$ .\n\n$\\therefore \\triangle N G K \\sim \\triangle G H C$ .\n\n$\\therefore \\angle K N G=\\angle C G H$ .\n\n由前知, $G K \\perp K E$ .因此, $E 、 K 、 G 、 N$ 四点共圆.\n\n$\\therefore \\angle C G H=\\angle K N G=\\angle G E K$ .\n\n$\\therefore \\angle E G C=\\angle E G K+\\angle C G H=\\angle E G K+\\angle G E K=90^{\\circ}$ .\n\n$\\therefore G E \\perp G C$ .']			['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAgcB7gMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKzdZ1iz0DSbzVdQmENpapvkc9/QD1JJAA7kgdaANKiqek6jDrGjWOp26usN5bx3EYcAMFdQwzgnnB9auUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRUcskcSF5XVFHJZjgD8TQFmSUVzt/438N6aD9o1i3Zv7kB80/kmcfjiucvPjDoUKn7La3ly/b5VRT+JOf0rKVanHdmE8TRh8Ul956LRXi978ZdTkJ+xabaQL0HnFpW/QqP0rIm+KfiqT7t3DDn/nnbr/UGsXjaS8/Q55ZlQXW/oe/0V83t4z8WXhbGrXrHv5Xyf+g1Xa58VX3LTazOD3LSvUfXov4Ysj+0ov4Ytn0xRXzOdK8Uyg/6BrDg8cwSn+lB0PxQhBGl6wp9reUf0pfXZfyMX9oy/wCfbPpiivmf7H4ot23G31mMjvslXFSpr/i2z4GoasmP4XkfA/On9eS3ix/2kl8UGj6Tor52t/iN4steBqzOPSWJHP5kZrTt/i74khx5iWU4HUPEQfzBH8qax9J73KjmVB73R7tRXlFj8aIiFGoaQ6+r28ufyBH9a6Gy+Kfhe7cLJcz2pJ48+E4/Ncit44ilLaX36HRHF0ZbSR21FULPWdN1AD7FqFrcEjpFMrH8s5q/2rbdaHQrS2CiikJ/LvmmGoyWRIY3kkYKiAszHgADkmvmL40eMdR8Qz2ccJaHw7LukswRg3O07TKRn7uchfbnvXo/jXxZp+u62PDTrqMugwlv7UnsLeSTzpFxi3DIpwAcb+R/d4Oa8y+OGvWGvaro50+3uoIre1aPZcWrQYG4YCqwHH0pAfQ/gX/knvhr/sFWv/opa6CsDwL/AMk98Nf9gq1/9FLW/QAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFALXYKKxdb8VaN4fU/wBoX0ccmM+Up3SH6KPX3xXmWvfF69uQ0Oi2wtIzx50wDSfgPuj8d1Y1MRTp7s56uKpUtJvU9fury2soTNdTxQRDq8rhR+Zri9Y+K2g6cTFZibUJR3hGEH1Y/wBAa8nhsvE/jK680Jeai6nHmSE7U9snhfXiuy0n4NXblH1bUY416tHbgsx9sngfka5vrNap/Cjp5nIsVXq/wY6d2Zup/FzXrt2FitvYxfw7U8xvxY5B/ACucMXiXxXcLJs1DUX6BipdV/PgfpXuOl+AvDelBTFpcUsi/wDLS4/eMT688D8BXSqAqhQMADAFP6rUnrUkH1GrU1qzZ4Zp/wAI/EN0oa7e1s1PVXfc3/joI/Wulsfgzp8YU6hqdxMe6wqqD8zk/wAq9PorWODox6XN4YChHpc46z+GPha1IZrB529ZpmP6AgfpW3aeGtDsQPs+kWUZH8QgXP54rWorZU4LZHRGlCPwpIRQFUKAAAMADtS0UVdjQKKKKAuFFFFC0ArXNjaXgxdWsE6+ksYb+YrFu/Afhe8B83RbZS3eIGP/ANBIro6KTjF7omUIS+JXPP7r4ReHZ1PkPeW5PTbKGA/76BP61zt98GLtWJsNXhkHZZ4yhH4gn+VexUVjLC0ZdDnngqE94nznqfgHxPpJMj6bJMi8iS1PmD68fMPxAqKH4g+J/DVtJi+mdYVJ8m6Xf06DnkD6Yr6Rrx746eK9MsdEOgrDb3OqXa8l1DNbR5+9zyCeQPQZPpmaWXSlUSot3MVgFCSlCTOOf4k+OZ5hef21HAThhapbR+UB/d+YFj9Sa6q8+Ly63o+i6bbSnTdR1OU299dKM/YgrBXKZ7t/CewPPIrw6PxBPHaCEwB3C7RIzdR9PWu28P8Ag/V9R8KWur6db/boJWbzRCQZFfdggr1zn0zwRVTo4zDRcqsbq57+c18BClT+oq8vtH0lo2nWOj6LaWOmqq2UMSrFtbcCOuc985yT36187fHnUIvEHijT49LV7uO2tdrzQLvTczE4BHUgAfnWhpPjvxL4emWI3ck0cfytb3aswHbHOGX8x9K9K0L4r6NqO2LUUfTp+BlzujJ/3gOPxAqaeLpz0ejPDo46lP3ZaPzNT4dX0F/8PtDETc21lDbzIRho5EjVWUjt0z9CK6ztUUM8NzEk0EqSROMq6MGBHsRUvaunY7QooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoqC5uoLS3knuJUiiQZZ3baFHua8s8V/Fna0tl4fVWGMG8ccH/AHR/U/lWdStGkryZjVxFOkryZ6FrniTSfD0Pm6ldpESCUiHLv9F6n+VeS+IPivq2pFodKT+z4M4EgIMrD6/w/h+dYGkeHNe8Y3vmxpPOGbEt3OxK++WbqfYZ/CvXPDfw10bQ8T3C/b7v/npOoKqf9lOg+pya4+etiPg0Rxc+IxWkNInlWieCfEPiiYXRikjhlO9ry5J+b3B5Zs+2frXp2h/CrQtLUPfBtRnGDulG1AfZQf5k13ajCge1LW9PCQhrLVm9HA0qerV35kcMMcESxQxpHGowEUYA+gqSiiuk7AooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKr3dzBZ2s1zcyrHBCjSSO54VRySfYCmt9AMTxp4tsvBvh+41S7ZTIAUt4S2DLIRwv04yT2Ga+YND0rV/id46Imkd7i7kMt5cBeIo+MnHYAYAH0FTePfGF78QPFgMAka0WTybC2HXBOAdv8AeY4/Qdq+hvht4Gg8EeHEhdFOpXAEl5L1+bHCA/3V/UknvXtRtgaHN9tmfxM+YvGWmW+jeMdV0yzRlt7W4MUYJycDAyffivWf2d9dXOraDI+C228hGevRH/8AZK81+IUYl+J2uRkn5r9wfzx/hT/BGrP4K+JFpLd/IkFy1rd5PCqco2f93734V3V4e2wvK97XJWj0PqrV/DWj66v/ABMrCKZ8YWTGHX6MOa8y8QfCCeBJJ9CuDOASRbT4DAezdD9CB9TXslFfIVaFOp8SFVw1KqveR822WseI/BV6YEkuLNwQz20y5Vh/unjn1A/GvTfDvxW03UiLfVlGn3B4EgJMTH69VP1/Ou21PR9P1m2NvqFpFcR848xclfcHt+FeUeJ/hLc2ga60GRrqPOTbSH5wPYn736H61y+yq4f+G7o4fZYjDa03ddj2RHWSNXRgyMAVYHII9adXzjoPi7XPCF28AaQxI2JbO4zgHoeOqH6Y9817P4X8a6X4ngUQyiG925e1kb5h64/vD6fjXRSxMKmj0kddDFwraPRnT0UDpRXQdXWwUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRSEgcnp1oAWub8UeMdM8LQf6UxluZM+VbRn5mHqf7o9z+Gelc141+JsOl+Zp+iOs16DiSfGUi+n94/oPfpXnWgeFta8aajJMrMU3ZnvJySM/j94/T8fWuOtiXf2dPVnBXxlpezpay/ITWvEeueNtTjhId9zYgs4ASg98Dkn1J/lXdeFPhOkBjvdfYSSKdy2SnKj03sOv0HH16V23hvwjpfhi322UW6dhiS4kwXf8ew9q3x0opYWz5qmrFRwWvPX1kRQwxW8SxwxLHGowqIoAH5VL2oorsPQCiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKADpXzz8cPiB9tun8LaZMfs9u+b50biSQf8s/ovUj1wO1d78W/iAnhLRGsLC4VdavFxEOphToZD6eg9/oa8e+E/gB/GWum+1COT+yLNw8xb/ltJ1EeT78n244yDXq4KhGnH6zV2WxEnfQ734I/Dz7Fbp4r1WNTPMn+gxsM+Wh6yexI6egz617bSIqoiooCqowABgAUtcOIryrTc57/oXFWPj34hcfFLW8f8/wCf51u/HHw+NH8eveRKBBqcYuPQLJ91vxyAf+BVg/ET/kqGu/8AX81e2fHvQhqHgqLVUjzNptwGLD/nm+Fb/wAe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fgfWvQy3Eeyq8r2ZMldHhHw91xfD3jzSNQdykAmEcxJ4Ebgq2foGz+FfZI6V8H/rnjP+fwr7H+H2vL4k8D6VqJfdKYVin9fMX5Wz9SCfxrszik9KiJgzqa4r4meIf7F8MyW8UgF1fZhjHcLj5j+Rx/wIV2ZIAyeB1zXzr4x1qTxZ4ud7bLwlhb2qnkEZwD+JJP4181iqvJCy3Zz42t7OnZbs3vhJ4f8AtusS6xPHmGyG2MnoZCP5gfzFe3DoKxfDGhReHdAt9Nj5dBmVz/G55J+meK2quhT9nBIvC0fZUkgooorY6QooooAKKKKAMnxDokPiDRLvTpwB5y4V8Z2sPukfQ/19a8F0XUrzwV4vDXCspglMN1GvJdO+P5j6CvpCvKviz4UM8Y8Q2i/NGAl2oHVf4X/Dofw9K48VTdvaR3RwY6lJpVYbo9Rt54rm3jnhcPFIoZGXowIyCKyvE3iGw8K6Dc6vqLlYIRwq/ekY9FX1JP8A9fgGvPfhV4t2pJoV/LsCBpLZ3OAoHLL/AFH0NeV/FLx7L4419bayDDSrNylsg585icFyO+eg9B7mvUy6j9cs1t1N6WIjUpKSZlf8Tz4o+PD/AMtLy9fHGdlvEP5BR9M+5NfVXhrw5YeFtEg0nToysEIOWbq7Hksfc/pXK/CnwAng7QRdXaA6veoGnY9Yl6iMfTv6n2Ar0SujH4pVZKnD4UbRjbcB0ooorzygooooAKKKKACiiigAooooAKKKKP66gFFFFH9dQCiiigAoooos+wBRRRTswCiiiizAKKKKLAFFFFIAooooAKKKKTaDY+U/jd/yVLUf+uUP/ota+mPDX/Iq6P8A9eMP/oAr5n+N+B8Ur/nrFB/6AtfS/hnnwpo//XjD/wCgCvWxtnh6diI7s1aKKK8osKKKKACiiigAooopXAKKKKd0AUx0WRGR1DKwwVPQin0UXXcD4/8AiV4TPhDxldWkSbbGc+faEdPLPb/gOCK9H/Z311QNW0CV8N8t5Cp/74f+SfrXa/GDwcvinwjLcwRltR00NPb7RkuuPnT8QPzA9a+ePAXidfCPjGx1aRWaBC0cyjjKMMH8s5+or31WjicG7vVGT913Pon4qeJTpeijSraUrd3oO7Z1WIdfz6fTNcz8J/DDXeotrt1GDb2522+R96X1HsB+p9q5PdqHjzxaShVp7yThgflijH/soH513PxL8WRfDXwXZ6Fokgj1G4Qxwt3iQffkIPck8e5JH3cV8jTi69TneyOCivrNZ1JbI63xP8R/DPhKdbbUL0vetjFpbL5kvtkDpntkiqY8dastoNRfwLryacVL+ZuhM4XB5MO/cD7VwPwE8HLcR3HjTVQ091NKyWjS/M2c/vJcnqxOVz7N617tjIxx/Su89Q4/wh8RNI8cXl5Bo0N2yWkSvNLNGEUFsgKBknPDdcdO/bsq5rwx4Tt/DV7r11EyFtVv2uvlGNikD5f++i5/4FXS0AFFFFABRUbqsisjqGRgQykZBHTFfInxF06xu/i1eaPodjBaxG4htY4rePaDIQqthR/tEjAoA+v6huIY7iGWCdFkikUo6MOCpGDn2xxXz38ZfBHh/wAG6BpOoaDC9jqAuRBujlYNMuxiWPPUEAZGPvc9q7r4HeIdY8Q+Cpn1ieS5e3umhhuZiWd12hvmJ5bBOM/h2oE+x4neafBc6zqayq7RQ3k0EMT5+RVcgZHrgVmW19J4R8TQahp8MEssQ8yJLlPMWNs9QM846g+v0r6B8WfCK01nU7nVdJ1GTS7u43PNEIvMilkP8eCcqx7469cZ6+R2OnReHvELDWtOjv5YXK3EN3GCCOnA6D1B71z0Kzwdfnn8L3sejjcyy6nl8KEaXLNPexY/4Xz40/vaf/4D/wD16P8AhfPjT+9Yf+A3/wBevbtO8FeA9W0+C+s/D+mywTruVvKH0I+o7irv/CvPB3/QtaZ/4DrXurGYNq6pXPLSk1ds8E/4Xz40/vWH/gN/9emH47eNiT+/sh7fZhXv3/CvPB3/AELOmf8AgOtH/CvPB3/Qs6Z/4DrT+t4P/nz+AWfc8B/4Xr42/wCfiy/8BhR/wvXxt/z8WX/gMK9+/wCFeeDv+hZ0z/wHWnL8PvB4HHhnSvxtUP8ASj63hP8AnyOz7nz/AP8AC9fG3/PxZf8AgMKQ/HPxuRxc2i/S1WvoP/hX/g//AKFjSf8AwET/AAo/4V/4P/6FjSf/AAET/ChYzCL/AJdByvufPX/C8vHH/P5a/wDgKlH/AAvLxx/z+2v/AICpX0L/AMK/8H/9CxpP/gIn+FA8A+EF6eGNI/GzQ/0qvr2F/wCfX4L/ACFyvufPX/C8vHH/AD+2v/gKlIfjj435H222B9Rap/hX0P8A8IH4R/6FjR//AACj/wAKX/hBPCP/AELGj/8AgFH/AIUfXsL/AM+vwX+Qcr7nzp/wu7x3/wBBOD/wEj/wo/4Xd47/AOgnB/4CR/4V9F/8IH4R/wChY0f/AMAo/wDCj/hBPCX/AELGj/8AgFH/AIUvr2F/59f19wcr7nzp/wALu8d/9BOD/wABI/8ACj/hd3jv/oJwf+Akf+FfRv8AwhHhIDH/AAi+i/8AgBF/8TS/8IR4S/6FfRP/AAXxf/E0fXsL/wA+v6+4OV9z5tb41+PSeNXjH/bpD/8AE0f8Lq8e/wDQZT/wEh/+Jr6S/wCEI8Jf9Cvon/gvi/8AiaP+EI8Jf9Cvon/gvi/+Jo+vYX/nyvu/4Acr7nzb/wALq8e/9BlP/ASH/wCJo/4XV49/6DKf+AkP/wATX0mvgzwsv3fDWjL9LGL/AOJp3/CHeGP+hc0j/wAAYv8A4mj69hf+fK+4OV9z5ob40ePmHGtqv0s4P6pSf8Lm8f8A/Qf/APJOD/4ivpj/AIQ7wx/0Lmkf+AMX/wATR/wh3hj/AKFzSP8AwBi/+Jo+vYX/AJ8r7h8r7nzN/wALm8f/APQf/wDJOD/4il/4XN4//wCg/wD+ScH/AMRX0wPCHhpTlfDukg+osox/7LUn/CK+Hv8AoA6X/wCAcf8AhR9ewn/PlC5X3PmI/GXx+R/yH/ytIP8A4im/8Li8ff8AQwN/4DQ//EV9P/8ACK+Hv+gDpf8A4Bx/4Uf8Ir4e/wCgDpf/AIBx/wCFH17C/wDPpfh/kHK+58wf8Li8ff8AQwN/4Cw//EUf8Li8ff8AQfb/AMBYf/iK+n/+EV8Pf9AHS/8AwDj/AMKkHhzQwABo2nADsLVP8KPr+G/59L8P8g5X3Plz/hcXj7/oPt/4Cw//ABFNPxe8esc/8JBJ/wCA8X/xFfUv/CO6H/0BtP8A/AVP8KP+Ed0P/oDaf/4Cp/hR9fw3/Ppfh/kHK+58tf8AC3vHn/QwS/8AfiL/AOIpD8XvHn/QwS9P+eEX/wARX1N/wjuh/wDQG0//AMBU/wAKVdA0Zc40iwGf+nZP8KTx+Gs0qS/D/IfK+58pRt/bSnU9SZby9uvnmmlALMT/ACAAwAPSlh+IfizQFbStN16eOyt2CxRlVfYOu0Myk4HTFe8ax8G/D2qanLe29xf6aJmLSwWciqjN3IDKdue+OPaun0rwd4f0jTo7K00m1ESAjMkYdmPclmGSTXkYROjWdSfvR7P+me1j8ww+Jw0KVOnyyW77nzH/AMLZ8d/9DDN/36j/APiaP+FteO/+hhn/AO/Uf/xNfVf9haR/0CrH/wAB0/wpRomkqcrpdkCO4gUf0r1/r+H/AOfK/D/5E8Xlfc+U/wDhbXjr/oYZ/wDv1H/8TSH4teOsY/4SGf8A79R//E19Yf2Vp3/Pha/9+V/wo/srTv8Anwtf+/K/4UfX6H/Plfh/8iHK+58m/wDC1vHP/QxXP/fKf/E0f8LW8c/9DFc/98p/8TX1l/ZWnf8APha/9+V/wo/srTv+fC1/78r/AIUf2hR/58r8P/kQ5X3Pk3/ha3jn/oYrn/vlP/iaP+FreOf+hiuf++U/+Jr61FhZgAC1gAHbyxR9htP+fWD/AL9ij+0KP/Plfh/8iHK+58kH4p+OCc/8JHd/kv8AhR/wtLxv/wBDHd/+O/4V9b/YbT/n1g/79ij7Daf8+sH/AH7FH9o0V/y5X4f5C5H3Pkj/AIWl43/6GO7/APHf8KP+FpeN/wDoY7v/AMd/wr65S0t0+5bxL9EAp/kQ/wDPJP8AvkU/7Rpf8+V9/wDwB8kujPkI/FDxs3B8R3ZHcfL+R4qvo0EE1k1wypLPKzGRmAPzZr3/AOJfi+LSLc6Np/lC9nX99IODAn4fxH9OvpXCeBfhlb+KZbjULyS8srJcKDav5Zmf6kdvb25ryMxxsMU1RpR5bb2OvKc2o4LGe/DnVu2xs/Aq0hTUvEbxwrsj8lUkA+4WDF1Hp91T+Vee/HueWX4nzrKflitYUjHouNxHvyxr6W0Hw3pXhnTfsGk2i28G7c2MsXYjBZieWOPWvOvjF8L5/FkSa5oy79Yt0ET25YATxgk8Z43DJ69Rx1xUwhypJk4ipGrVlOCsm72O/wDBmmxaT4K0WxiC7IrOIEqMBmKgs2PUsSfxreHSvGfBXxLv9F0iDQ/F/h7W4L60jEcc6WbuJkHyqSOu7jGeQcZyK7S21rXfEtz5VjpV5oumg/vr7UYvLnkUHlY4icqTz87dOoBqjE7KikUbVAGeBjk5paACiiigCtf3kWn6fc3sxxFbxNK59FUEn9BXyd8OLDW/FvxHl1eyi0+a+t3fUpBfs6xGQvwfkBO4MwYf7v4V7h8cdeOi/Di5hjbE2oyLaKfRSCzn/vlSv/Aq539nXQWtPDeo65KOb6ZYogcfcjzyPqzEf8AoA43xHqMuqeOo9P8Ai1LeWUMI32yaci/Z0ViMnoWKHafmBZsjHGK+ifD+n6TpehWdvocUMemCMPAIW3KysN27dyWznOSTnNfNfxs8Rw+KPHkFhpX+kpZR/ZVaIbzJMW5Vcfe/hAHXIPtX0J4B0i70HwJo2mXwxdQWwEgz91iSdv4Zx+FGnUDpa8++JPgr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']	Multimodal	Competition	False				Theorem proof	Plane Geometry	Math	Chinese