Question 22
Let $H$ and $O$, respectively, be the orthocenter and circumcenter of the acute triangle $ABC$ ($AB \neq AC$); Let $\Gamma$ be the circumference circumscribed by the triangle $ABC$. The extension of the median $AM$, relative to the side $BC$, cuts $\Gamma$ at the point $N$; diameter circle $AM$ cuts $\Gamma$ at points $A$ and $P$.
Prove that the lines $AP$, $BC$, and $OH$ are concurrent if, and only if, $AH = HN$.
Prove that the lines $AP$, $BC$, and $OH$ are concurrent if, and only if, $AH = HN$.