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 {$S_i$}.


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.

  • $\begingroup$ Do you mean that $C_1$ has its centre on the real axis? $\endgroup$ – user441558 Aug 27 '17 at 10:27
  • $\begingroup$ It's a valid assumption, isn't it? $\endgroup$ – scd Aug 27 '17 at 10:36
  • $\begingroup$ Yup, of course, because translation is a conformal mapping. $\endgroup$ – user441558 Aug 27 '17 at 10:38

Use an inversia such that two given circles will be with common center.

  • $\begingroup$ Sorry, I don't quite get what do you mean? $\endgroup$ – scd Aug 27 '17 at 10:40
  • $\begingroup$ @KIYORI If for given two circles there is a common center then your statement is obvious. About inversia see here: en.wikipedia.org/wiki/Inversive_geometry $\endgroup$ – Michael Rozenberg Aug 27 '17 at 10:44
  • $\begingroup$ Do you mean that if C1 and C2 are not co-centered, we can take inversion to make them co-centered? $\endgroup$ – scd Aug 27 '17 at 11:23
  • $\begingroup$ That's correct. $\endgroup$ – John Hughes Aug 27 '17 at 11:24
  • $\begingroup$ @KIYORI Y Yes of course. I am ready to explain how we can find the circle of the inversia. $\endgroup$ – Michael Rozenberg Aug 27 '17 at 11:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.