Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 14F (Geometry)

Cross-ratio conventions differ by permutations. Use the following ordering, which gives the signs requested in part (c):Values involving infinity are defined by limits. This is the cross-ratio with second point mapped to zero: the Möbius transformationsends $z_1,z_2,z_4$ to $1,0,\infty$ and sends $z_3$ to $r$.

For $M(z)=(az+b)/(cz+d)$, $ad-bc\ne 0$, the identityshows that all factors cancel in the cross-ratio. This proves Möbius invariance of the cross-ratio, with the cases involving infinity following by limits. Algebraic subtraction also givesThe latter expression is another common ordering for the cross-ratio; keeping the convention explicit prevents an incorrect reciprocal or sign.

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.

Write $z_j=c+R{e}^{i{\theta }_j}$ with ${\theta }_1<{\theta }_2<{\theta }_3<{\theta }_4<{\theta }_1+2\pi$. In the identitythe phase factors cancel in the cross-ratio. In our convention the two numerator sine factors are negative, while the denominator factors have opposite signs. Thus $r<0$, and the exchanged cross-ratio $1-r$ is greater than one. Consequently $|r|+1=1-r=|1-r|$.

Taking absolute values of the two formulas in part (a) givesMultiplying $|r|+1=|1-r|$ by the common positive denominator proves Ptolemy theorem:The product of the diagonals equals the sum of the products of opposite sides.