Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 14F / b / Solution

With $T$ from part (a), the first, second and fourth points map to $1,0,\infty$. The inverse image of the extended real line under a Möbius transformation is a generalized circle, meaning a circle or a line together with infinity. Thus, if $r=T(z_3)$ is real, all four original points lie on that generalized circle.

Conversely, if the four points lie on one generalized circle, its image under $T$ is another generalized circle containing $0,1,\infty$. A generalized circle containing infinity is a line, and the line through zero and one is the real line. Therefore $T(z_3)=r$ is real. Hence the cross-ratio is real exactly when the four points are concyclic or collinear.