Finding a third point I have learnt that if we are given 3 points in the extended complex plane and their corresponding image points, we have a unique Möbius map that can perform the mapping. Suppose I have 2 orthogonally intersecting circles and I want to map them (individually) to the real and imaginary axes respectively by some Möbius map, is there a systematic way to do so? I have figured that the intersection points will have to be sent to $0$ and $\infty$ respectively but how might I determine a third point and its image so as to define such a map?

Let one of the intersections be $p$.  The inversion $z \to 1/(z-p)$ takes $p$ to $\infty$ and takes your two circles to straight lines intersecting orthogonally.  Now just translate and rotate.  
