BdMO Online Forum

4. 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 $w_1$ is a circle passing through $D$ and tangent to $AC$ at $P$. Also suppose that $w_2$ is a circle passing through $C$ and tangent to $BD$ at $P$. Let $X$ be the intersection point of $w_1$ and $AD$, and $Y$ be the intersection point of $w_2$ and $BC$. Suppose that the circles $w_1$ and $w_2$ 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 $XQY$ .