Question 3
In the plane, let $ABCD$ be a convex quadrilateral inscribed in a circle with center $O$, with neither pair of opposite sides parallel. Let $M=AB\cap CD$ and $N=AD\cap BC$, where the intersections are of the supporting lines. Let $P,Q,S,T$ respectively be the intersections of the internal angle bisectors of the pairs $\angle MAN$ and $\angle MBN$; $\angle MBN$ and $\angle MCN$; $\angle MCN$ and $\angle MDN$; $\angle MDN$ and $\angle MAN$. Assume that $P,Q,S,T$ are pairwise distinct.
(1) Prove that $P,Q,S,T$ lie on one circle, and let $I$ be its center.
(2) Let $E=AC\cap BD$. Prove that $E,O,I$ are collinear.
(1) Prove that $P,Q,S,T$ lie on one circle, and let $I$ be its center.
(2) Let $E=AC\cap BD$. Prove that $E,O,I$ are collinear.