Question 135
In triangle $ABC$, $BC \gt AB$. Let $L$ be the internal angle bisector of $\angle ABC$. Drop perpendiculars from $A$ and $C$ to $L$, with feet $P$ and $Q$ respectively. Let $M, N$ be the midpoints of sides $AC$ and $BC$, respectively. Let $O$ be the center of the circumcircle of triangle $PQM$, and let $H$ be the second intersection of this circle with $AC$. Prove that $O, M, N, H$ are concyclic.