MathNet 2016-part-06 — 184 problems (EN–VI) Question 129

Let $ABQP$ be a convex quadrilateral. Let the angle bisectors of $\angle PAQ$ and $\angle PBQ$ intersect at $C$. The circumcircle of $APQ$ intersects $AB$, $AC$ at $M$, $N \ne A$, respectively. Lines $PQ$ and $BC$ intersect at $S$. Prove that if $AC$ is perpendicular to $BC$ then $M$, $S$, $N$ are collinear.
*(Chesphongphach Buranasilp, Krit Boonsiriseth and Pachara Sawettamalya)*