KöMaL 2018 B4949

The feet of the altitudes drawn from vertices $B$ and $C$ of an acute-angled triangle $ABC$ are $D$ and $E$, respectively. Let $P$ be an interior point of $AD$, and let $Q$ be an interior point of $AE$ such that $EDPQ$ is a cyclic quadrilateral. Show that the line segments $BP$ and $CQ$ intersect on the median drawn from $A$.