KöMaL 2018 A716

Let $ABC$ be a triangle and let $D$ be a point in the interior of the triangle which lies on the angle bisector of $\angle BAC$. Suppose that lines $BD$ and $AC$ meet at $E$, and that lines $CD$ and $AB$ meet at $F$. The circumcircle of $ABC$ intersects line $EF$ at points $P$ and $Q$. Show that if $O$ is the circumcenter of $DPQ$, then $OD$ is perpendicular to $BC$.