Question 137
Let $ABC$ be an acute-angled triangle. The altitudes $BE$, $CF$ meet at $H$. $O$ is the circumcenter of triangle $ABC$. $M$ is the midpoint of $BC$. $P$ is the point on $EF$ such that $HP \perp HO$. $Q$ is the point on $HA$ such that $PQ \perp HM$. Prove that $QA = 3QH$.