11th European Girls' Mathematical Olympiad | EGMO 2022

Problem 6. Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ . Let the internal angle bisectors at $A$ and $B$ meet at $X$ , the internal angle bisectors at $B$ and $C$ meet at $Y$ , the internal angle bisectors at $C$ and $D$ meet at $Z$ , and the internal angle bisectors at $D$ and $A$ meet at $W$ . Further, let $AC$ and $BD$ meet at $P$ . Suppose that the points $X$ , $Y$ , $Z$ , $W$ , $O$ , and $P$ are distinct.
Prove that $O$ , $X$ , $Y$ $Z$ , $W$ lie on the same circle if and only if $P$ , $X$ , $Y$ , $Z$ , and $W$ lie on the same circle.