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For any two circles there are two homothety centers, and I have to show this are the only possible centers of homothety . I started by supposing that a point Q is a center of homothety. Then, If we trace a tangent through Q, of one of the circles, then, it is tangent to the other circle, since a homothety is a one-to-one correspondence. Therefore, Q must be any of the 4 intersections between the four common tangents to the circles. Therefore, I only have to prove that the intersections of the exterior tangents with the internal ones, are not homothety centers. I think my approach is not very subtle, and will not lead to much, and I can’t see how to prove they are unique. Thanks.

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