2010 APMO 第 4 题

Let $A B C$ be an acute triangle satisfying the condition $A B>B C$ and $A C>B C$. Denote by $O$ and $H$ the circumcenter and the orthocenter, respectively, of the triangle $A B C$. Suppose that the circumcircle of the triangle $A H C$ intersects the line $A B$ at $M$ different from $A$, and that the circumcircle of the triangle $A H B$ intersects the line $A C$ at $N$ different from $A$. Prove that the circumcenter of the triangle $M N H$ lies on the line $O H$.

令$A B C$ 为锐角三角形，满足条件$A B>B C$ 且$A C>B C$。 $O$ 和 $H$ 分别表示三角形 $A B C$ 的外心和垂心。假设三角形$A H C$的外接圆与线$A B$相交于与$A$不同的$M$处，并且三角形$A H B$的外接圆与线$A C$相交于与$A$不同的$N$处。证明三角形 $M N H$ 的外心位于直线 $O H$ 上。