IGO 2016 A4

In a convex quadrilateral $ABCD$ , the lines $AB$ and $CD$ meet at point $E$ and the lines $AD$ and $BC$ meet at point $F$ . Let $P$ be the intersection point of diagonals $AC$ and $BD$ . Suppose that $\omega_1$ is a circle passing through $D$ and tangent to $AC$ at $P$ . Also suppose that $\omega_2$ is a circle passing through $C$ and tangent to $BD$ at $P$ . Let $X$ be the intersection point of $\omega_1$ and $AD$ , and $Y$ be the intersection point of $\omega_2$ and $BC$ . Suppose that the circles $\omega_1$ and $\omega_2$ intersect each other in $Q$ for the second time. Prove that the perpendicular from $P$ to the line $EF$ passes through the circumcenter of triangle $XQY$ . Proposed by Iman Maghsoudi