OMCC 2019 Problem 4

Let $ABC$ be a triangle, $\Gamma$ its circumcircle and $l$ the tangent to $\Gamma$ through $A$ . The altitudes from $B$ and $C$ are extended and meet $l$ at $D$ and $E$ , respectively. The lines $DC$ and $EB$ meet $\Gamma$ again at $P$ and $Q$ , respectively. Show that the triangle $APQ$ is isosceles.