OMM 1989 Problem 5

Let $C_1$ and $C_2$ be two tangent unit circles inside a circle $C$ of radius $2$ . Circle $C_3$ inside $C$ is tangent to the circles $C,C_1,C_2$ , and circle $C_4$ inside $C$ is tangent to $C,C_1,C_3$ . Prove that the centers of $C,C_1,C_3$ and $C_4$ are vertices of a rectangle.