ILL 1986 Problem 49

Let $C_1, C_2$ be circles of radius $1/2$ tangent to each other and both tangent internally to a circle $C$ of radius $1$ . The circles $C_1$ and $C_2$ are the first two terms of an infinite sequence of distinct circles $C_n$ defined as follows: $C_{n+2}$ is tangent externally to $C_n$ and $C_{n+1}$ and internally to $C$ . Show that the radius of each $C_n$ is the reciprocal of an integer.