Past exam of the short options course of the University of Oxford / 2015 / 3 / d / Solution

By invariance under projective transformations and part (b), it is enough to calculate on the normalized quadruple $(\infty ,0,1,\lambda )$. Direct evaluation of the cross-ratio givesFor instance, $(0\infty :1\lambda )=(0-1)/(0-\lambda )=1/\lambda$, and $(\infty 1:0\lambda )=(1-\lambda )/(1-0)$. ThusAll denominators are legitimate because $\lambda \notin \{0,1\}$. The first identity explains why some distinct permutations always give the same cross-ratio.