2000 APMO 第 3 题

Let $A B C$ be a triangle. Let $M$ and $N$ be the points in which the median and angle bisector, respectively, at $A$ meet the side $B C$. Let $Q$ and $P$ be the points in which the perpendicular at $N$ to $N A$ meets $M A$ and $B A$, respectively, and $O$ be the point in which the perpendicular at $P$ to $B A$ meets $A N$ produced. Prove that $Q O$ is perpendicular to $B C$.

设 $A B C$ 为三角形。令 $M$ 和 $N$ 分别为 $A$ 处的中线和平分线与边 $B C$ 相交的点。设$Q$和$P$分别为$N$到$N A$的垂线与$M A$和$B A$相交的点，$O$为$P$到$B A$的垂线与$A N$相交的点。证明$Q O$ 垂直于$B C$。