ISL 1979 Problem 51

Let $ABC$ be an arbitrary triangle and let $S_1, S_2,\cdots, S_7$ be circles satisfying the following conditions: $S_1$ is tangent to $CA$ and $AB$ , $S_2$ is tangent to $S_1, AB$ , and $BC,$ $S_3$ is tangent to $S_2, BC$ , and $CA,$ .............................. $S_7$ is tangent to $S_6, CA$ and $AB.$ Prove that the circles $S_1$ and $S_7$ coincide.