2008 IMO Shortlist G7

Let $A B C D$ be a convex quadrilateral with $A B \neq B C$. Denote by $\omega_{1}$ and $\omega_{2}$ the incircles of triangles $A B C$ and $A D C$. Suppose that there exists a circle $\omega$ inscribed in angle $A B C$, tangent to the extensions of line segments $A D$ and $C D$. Prove that the common external tangents of $\omega_{1}$ and $\omega_{2}$ intersect on $\omega$.

设 $A B C D$ 为凸四边形，且 $A B \neq B C$。用 $\omega_{1}$ 和 $\omega_{2}$ 表示三角形 $A B C$ 和 $A D C$ 的内切圆。假设存在一个圆 $\omega$ 内切于角度 $A B C$，与线段 $A D$ 和 $C D$ 的延长线相切。证明$\omega_{1}$ 和$\omega_{2}$ 的公外切线相交于$\omega$。