Kanada 2011

Let $ABCD$ be a cyclic quadrilateral whose opposite sides are not parallel, $X$ the intersection of $AB$ and $CD$, and $Y$ the intersection of $AD$ and $BC$. Let the angle bisector of $\angle AXD$ intersect $AD$, $BC$ at $E$, $F$ respectively and let the angle bisector of $\angle AYB$ intersect $AB$, $CD$ at $G$, $H$ respectively. Prove that $EGFH$ is a parallelogram.