2012 APMO 第 1 题

Let $ABC$ be an acute triangle. Denote by $D$ the foot of the perpendicular line drawn from the point $A$ to the side $BC$ , by $M$ the midpoint of $BC$ , and by $H$ the orthocenter of $ABC$ . Let $E$ be the point of intersection of the circumcircle $\Gamma$ of the triangle $ABC$ and the half line $MH$ , and $F$ be the point of intersection (other than $E$ ) of the line $ED$ and the circle $\Gamma$ . Prove that $\tfrac{BF}{CF} = \tfrac{AB}{AC}$ must hold.

(Here we denote $XY$ the length of the line segment $XY$ .)

设$ABC$为锐角三角形。 $D$ 表示从点$A$ 到边$BC$ 所画垂直线的脚，$M$ 表示$BC$ 的中点，$H$ 表示$ABC$ 的垂心。设$E$为三角形$ABC$的外接圆$\Gamma$与半直线$MH$的交点，$F$为直线$ED$与圆$\Gamma$的交点($E$除外)。证明 $\tfrac{BF}{CF} = \tfrac{AB}{AC}$ 必须成立。

(这里我们将 $XY$ 表示线段 $XY$ 的长度。)