Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 2 / 12A / ii / Solution

Define the Möbius transformationThe three prescribed distinct points go to $0,1,\infty$, respectively, and the specified cross-ratio is exactly $T(z_4)$.

To identify the preimage of the extended real line directly, write $T(z)=(az+b)/(cz+d)$. The condition $T(z)=\overline{T(z)}$ becomesAfter division by $2i$, this has the real form $A|z{|}^2+Bz+\overline{B}\overline{z}+C=0$, with $A,C$ real. If $A=0$ it is a line; otherwise completing the square gives a circle. It contains $z_1,z_2,z_3$, including the pole as a point mapped to infinity, so it is nondegenerate. Three distinct noncollinear points determine one circle; if the three are collinear, their unique generalized circle is their line.

Consequently $T(z_4)$ is real exactly when the fourth point is on the same generalized circle as the first three. This proves the real cross-ratio criterion for a generalized circle: