In a triangle ABC, the incircle or inscribed circle of center O and the excircle or escribed circle of center O1 relative to AC are tangent to the sides at D, E, F, D1, E1, and F1, as shown in the figure. AO extended meets DE at A1, CO extended meets DE at C1. AO1 cuts D1E1 at A2. CO1 cuts D1E1 at C2. Prove that the points A, C1, A1, C, C2, and A2 are concyclic (lie on a circle).
Geometry problem 1271 in motion.