APMO 2010 - Asian Pacific Mathematics Olympiad Question 4

Let $ABC$ be an acute triangle with $AB\gt BC$ and $AC\gt BC$. Let $O,H$ be its circumcenter and orthocenter. The circumcircle of $AHC$ intersects line $AB$ again at $M\ne A$, and the circumcircle of $AHB$ intersects line $AC$ again at $N\ne A$. Prove that the circumcenter of triangle $MNH$ lies on line $OH$.

Newly drawn configuration: