Mobius Geometry identity

Question

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.

Answer 1

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, $b\mapsto1$.