ÖMO 2011 Problem 3

We are given a non-isosceles triangle $ABC$ with incenter $I$ . Show that the circumcircle $k$ of the triangle $AIB$ does not touch the lines $CA$ and $CB$ . Let $P$ be the second point of intersection of $k$ with $CA$ and let $Q$ be the second point of intersection of $k$ with $CB$ . Show that the four points $A$ , $B$ , $P$ and $Q$ (not necessarily in this order) are the vertices of a trapezoid.