MathNet 2009-part-01 — 175 problems (EN–VI) Question 148

Let $H$ be the orthocenter of an acute triangle $ABC$, and $D$ be the midpoint of the side $BC$. A line passing through the point $H$ meets the sides $AB, AC$ at the points $F, E$ respectively, such that $AE = AF$. The ray $DH$ meets the circumcircle of $\triangle ABC$ at the point $P$.

Prove that $P, A, E, F$ are concyclic.