CN-TST 2016 Problem 5

Refer to the diagram below. Let $ABCD$ be a cyclic quadrilateral with center $O$ . Let the internal angle bisectors of $\angle A$ and $\angle C$ intersect at $I$ and let those of $\angle B$ and $\angle D$ intersect at $J$ . Now extend $AB$ and $CD$ to intersect $IJ$ and $P$ and $R$ respectively and let $IJ$ intersect $BC$ and $DA$ at $Q$ and $S$ respectively. Let the midpoints of $PR$ and $QS$ be $M$ and $N$ respectively. Given that $O$ does not lie on the line $IJ$ , show that $OM$ and $ON$ are perpendicular.

Attachments: