IMAC 2009 Problem 2

In the triangle $ABC$ , the circle with the center at the point $O$ touches the pages $AB, BC$ and $CA$ in the points $C_1, A_1$ and $B_1$ , respectively. Lines $AO, BO$ and $CO$ cut the inscribed circle at points $A_2, B_2$ and $C_2,$ respectively. Prove that it is the area of the triangle $A_2B_2C_2$ is double from the surface of the hexagon $B_1A_2C_1B_2A_1C_2$ .

(Moldova)