I've come across the following problem in complex analysis:
Let $C_1$, $C_2$ be two circles (not generalized) having the same center (we call such circles concentric circles) and let $f$ be a Mobius transformation. Show that if $f(C_1)$, $f(C_2)$ are also concentric circles, then $f$ either preserves the ratio of their radii or inverts it.
I know that every Mobius transformation is a composition of trasformations of 4 simple types:
- Rotation
- Scaling
- Inversion
- Translation
It is fairly easy to show that all but inversion preserve their ratio of radii the same and furthermore preserve them concentric. It is also easy to show that inversion inverts the radii ratio when the center of the circles is located at the origin.
I really want to do a reduction from the general case where the center of the circles is in a general point at the phase of inversion to the case where they are centered at the center.
I can't find the right justification to do so. I also came across this Question: Möbius transformations and concentric circles which didn't provide a justification for why such reduction is correct. does anybody have a formal solution for this question?
Thank you!