Take a triangle with a circle $\Gamma_0$ tangent to two of three sides (you may also think that sides of the triangle are made out of circle arcs). Construct a chain of circles $\Gamma_1,\Gamma_2,\dots$ on such a way that $\Gamma_{n+1}\not=\Gamma_{n-1}$ is tangent to $\Gamma_n$ and two of the sides of triangle. Prove that $$\Gamma_6=\Gamma_0.$$

I do not know the proof, but I was told that it is hard to do without knowing elliptic functions.

P.S. I do not know the references --- please feel free to add it :)