Question 4
Let $ABC$ be an acute triangle satisfying the condition $AB \gt BC$ and $AC \gt BC$. Denote by $O$ and $H$ the circumcenter and the orthocenter, respectively, of the triangle $ABC$. Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and that the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A$. Prove that the circumcenter of the triangle $MNH$ lies on the line $OH$.