2014 IMO 第 4 题

Let $P$ and $Q$ be on segment $B C$ of an acute triangle $A B C$ such that $\angle P A B =$ $\angle B C A$ and $\angle C A Q = \angle A B C$ . Let $M$ and $N$ be points on $\overline{{A P}}$ and $\overline{{A Q}}$ , respectively, such that $P$ is the midpoint of $\overline{{A M}}$ and $Q$ is the midpoint of $\overline{{A N}}$ . Prove that $\overline{{B M}}$ and $\overline{{C N}}$ meet on the circumcircle of $\triangle A B C$

令 $P$ 和 $Q$ 位于锐角三角形 $A B C$ 的线段 $B C$ 上，使得 $\angle P A B =$ $\angle B C A$ 且 $\angle C A Q = \angle A B C$ 。设$M$和$N$分别为$\overline{{A P}}$和$\overline{{A Q}}$上的点，使得$P$是$\overline{{A M}}$的中点，$Q$是$\overline{{A N}}$的中点。证明$\overline{{B M}}$和$\overline{{C N}}$在$\三角形A B C$的外接圆上相交