ÖMO 2015 Problem 5

Let I be the incenter of triangle $ABC$ and let $k$ be a circle through the points $A$ and $B$ . The circle intersects

• the line $AI$ in points $A$ and $P$ * the line $BI$ in points $B$ and $Q$ * the line $AC$ in points $A$ and $R$ * the line $BC$ in points $B$ and $S$ with none of the points $A,B,P,Q,R$ and $S$ coinciding and such that $R$ and $S$ are interior points of the line segments $AC$ and $BC$ , respectively.

Prove that the lines $PS$ , $QR$ , and $CI$ meet in a single point.

(Stephan Wagner)