The acute triangle $ABC$ with $AB\neq AC$ has circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. The midpoint of $BC$ is $M$, and the extension of the median $AM$ intersects $\Gamma$ at $N$. The circle of diameter $AM$ intersects $\Gamma$ again at $A$ and $P$. Show that the lines $AP$, $BC$, and $OH$ are concurrent if and only if $AH = HN$.
Problem
Source: XV Rioplatense Mathematical Olympiad (2006), Level 3
Tags: geometry, circumcircle, parallelogram, geometric transformation, reflection, perpendicular bisector, BritishMathematicalOlympiad