IMO Shortlist 2005 problem G2

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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
Source: Međunarodna matematička olimpijada, shortlist 2005