Question 51
Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\omega$ having center $O$. The lines $AD$ and $BC$ intersect at $E$, and the lines $AB$ and $CD$ intersect at $F$. Let $P$ be a point on the segment $EF$ such that $OP \perp EF$. The circle $\Gamma_1$ passes through the points $A, E$ and is tangent to $\omega$ at $A$. Similarly, the circle $\Gamma_2$ passes through the points $C, F$ and is tangent to $\omega$ at $C$. If $\Gamma_1$ and $\Gamma_2$ intersect at $X$ and $Y$, prove that $PO$ bisects $\angle XPY$.