ELMO SL 2018 G1

Let $ABC$ be an acute triangle with orthocenter $H$ , and let $P$ be a point on the nine-point circle of $ABC$ . Lines $BH, CH$ meet the opposite sides $AC, AB$ at $E, F$ , respectively. Suppose that the circumcircles $(EHP), (FHP)$ intersect lines $CH, BH$ a second time at $Q,R$ , respectively. Show that as $P$ varies along the nine-point circle of $ABC$ , the line $QR$ passes through a fixed point.

Proposed by Brandon Wang