CMO 2016 Problem 5

Let $\triangle A B C$ be an acute-angled triangle with altitudes $A D$ and $B E$ meeting at $H$. Let $M$ be the midpoint of segment $A B$, and suppose that the circumcircles of $\triangle D E M$ and $\triangle A B H$ meet at points $P$ and $Q$ with $P$ on the same side of $C H$ as $A$. Prove that the lines $E D$, $P H$, and $M Q$ all pass through a single point on the circumcircle of $\triangle A B C$.