IMO 2005 Problem 1

Six points are chosen on the sides of an equilateral triangle $ABC$ : $A_1$ , $A_2$ on $BC$ , $B_1$ , $B_2$ on $CA$ and $C_1$ , $C_2$ on $AB$ , such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths.

Prove that the lines $A_1B_2$ , $B_1C_2$ and $C_1A_2$ are concurrent.

Bogdan Enescu, Romania