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.
