1991 IMO 第 2 题

(JAP 5) For an acute triangle $A B C, M$ is the midpoint of the segment $B C, P$ is a point on the segment $A M$ such that $P M=B M, H$ is the foot of the perpendicular line from $P$ to $B C, Q$ is the point of intersection of segment $A B$ and the line passing through $H$ that is perpendicular to $P B$, and finally, $R$ is the point of intersection of the segment $A C$ and the line passing through $H$ that is perpendicular to $P C$. Show that the circumcircle of $\triangle Q H R$ is tangent to the side $B C$ at point $H$.

(JAP 5) 对于锐角三角形 $A B C，M$ 是线段 $B C 的中点，P$ 是线段 $A M$ 上的一点，使得 $P M=B M，H$ 是从 $P$ 到 $B C 的垂线的底脚，Q$ 是线段 $A B$ 与穿过 $H$ 的垂直于 $P B$ 的直线的交点，最后，$R$ 是线段 $A C$ 与穿过 $H$ 且垂直于 $P C$ 的线。证明 $\三角形 Q H R$ 的外接圆在点 $H$ 处与边 $B C$ 相切。