Prove: For any given circles R and R' in C_oo, there is a mobius transformation T such that T(r)=R'. Further, we can specify that T take any 3 points on R onto any 3 points of R'. If we do specify Tz_j for j=2,3,4 (distinct z_j in R), then T is unique.
Thank you for clarifications.
Here is for reference the text of the theorem as you wrote it in the message:
Theorem: Suppose R and R' are 2 circles in C_oo. So, we know that there is a Mobius map T such that T(R) = R' (basically circles go to circles). If a,b,c are any 3 points on R and we can specify T(a), T(b), T(c), then such a T is uniquely determined. (Conway says that this proof is trivial, and we may use the fact that M-maps take circles onto circles).
The proof can perhaps be indeed regarded as trivial when you are aware of its two key elements:
A: If we take an infinite straight line as a case of a circle with infinite radius, then is can be proved that ANY Mobius transformation maps ANY ...
Mobius Transformations for Circles are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.