IMO 2008 Problem 6

Let $ABCD$ be a convex quadrilateral with $BA\neq BC$ . Denote the incircles of triangles $ABC$ and $ADC$ by $\omega_{1}$ and $\omega_{2}$ respectively. Suppose that there exists a circle $\omega$ tangent to ray $BA$ beyond $A$ and to the ray $BC$ beyond $C$ , which is also tangent to the lines $AD$ and $CD$ . Prove that the common external tangents to $\omega_{1}$ and $\omega_{2}$ intersect on $\omega$ .

Author: Vladimir Shmarov, Russia