2013 IMO 第 4 题

Let $A B C$ be an acute triangle with orthocenter $H$ , and let $W$ be a point on the side $\overline{{B C}}$ , between $B$ and $C$ . The points $M$ and $N$ are the feet of the altitudes drawn from $B$ and $C$ , respectively. Suppose $\omega_{1}$ is the circumcircle of triangle $B W N$ and $X$ is a point such that $\overline{{W X}}$ is a diameter of $\omega_{1}$ . Similarly, $\omega_{2}$ is the circumcircle of triangle $C W M$ and $Y$ is a point such that $\overline{{W Y}}$ is a diameter of $\omega_{2}$ . Show that the points $X$ , $Y$ , and $H$ are collinear.

设 $A B C$ 为锐角三角形，其垂心为 $H$ ，并设 $W$ 为边 $\overline{{B C}}$ 上位于 $B$ 和 $C$ 之间的点。点 $M$ 和 $N$ 分别是从 $B$ 和 $C$ 绘制的高度的英尺。假设 $\omega_{1}$ 是三角形 $B W N$ 的外接圆，$X$ 是一个点，使得 $\overline{{W X}}$ 是 $\omega_{1}$ 的直径。类似地，$\omega_{2}$ 是三角形 $C W M$ 的外接圆，$Y$ 是一个点，使得 $\overline{{W Y}}$ 是 $\omega_{2}$ 的直径。证明点 $X$ 、 $Y$ 和 $H$ 共线。