# Mobius Geometry identity

How do you prove that all figures consisting of three distinct points are congruent in Mobius Geometry?

I understand it relates to the Fundamental Theorem of Mobius Geometry. The concepts of which are very hard for me to grasp.

• For a (rather understandible) application to quantum computing, take a look to "symmetric entanglement classes for n qubits " by Martin Aulbach that you will find on ResearchGate – Jean Marie Jan 31 at 21:55

I suppose there are many ways of looking at this, but here’s how I think of it: take your three distinct complex numbers, $$\{a,b,c\}$$. Then you can get a Möbius transformation mapping these to $$\{0,1,\infty\}$$, namely $$z\mapsto f(z)=\frac{z-a}{z-c}\,,$$ except of course that this sent $$b$$ to $$\lambda=\frac{b-a}{b-c}$$ instead of to $$1$$. Well: just change that $$f$$ to $$\frac1\lambda f$$. You still have $$a\mapsto0$$, $$c\mapsto\infty$$, and now, besides, $$b\mapsto1$$.