SMO 2007

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

find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ st $f(f(m)+f(n))=m+n \,\forall m,n\in\mathbb{N}$ related: https://artofproblemsolving.com/community/c6h381298