KöMaL 2014 B4618

The polygon $A_1 A_2 \ldots A_n$ has both an inscribed circle and a circumscribed circle. The centre of the inscribed circle is $O$, and the centre of the circle $OA_i A_{i+1}$ is $C_i$ ($i=1, 2,\ldots, n$ , and $A_{n+1}$=$A_1$). Prove that $C_1$,$C_2$,...,$C_n$ are concyclic.