ZA-SAMO 2004 Problem 4

Let $A_1$ and $B_1$ be two points on the base $AB$ of isosceles triangle $ABC$ (with $\widehat{C}>60^\circ$ ) such that $\widehat{A_1CB_1}=\widehat{BAC}$ . A circle externally tangent to the circumcircle of triangle $\triangle A_1B_1C$ is tangent also to rays $CA$ and $CB$ at points $A_2$ and $B_2$ respectively. Prove that $A_2B_2=2AB$ .