Consider two circles $C_1$ and $C_2$. $C_2$ lies within $C_1$. Let $S_1...S_n$ be circles lies between $C_1$ and $C_2$, and the following were satisfied:
(i):Each circle $S_i$ is tangent to C1 and C2
(ii):Each $S_i$ is tangent to $S_{i+1}$, $S_n$ is tangent to $S_1$.
The problem is to prove that the number n is independent of the choice of the circles {$S_i$}.
Attempt:
WIthout losing generality, we can assume that the big circle $C_1$ passes through the origin and orthogonal to real axis, then under conformal map 1\z, it becomes a vertical line. The small circle $C_2$ is mapped to another circle lie on one side of the vertical line. The small circles $S_i$ should be tangent to both the line and the image of $C_2$. But I don't know how to prov that the number of small circles are finite and independent of choice. Any help or hint is much appreciated.
Use an inversia such that two given circles will be with common center.