Given distinct points $z_1, z_2, z_3, z_4 \in \mathbb{C}\cup\{\infty\}$, define the cross ratio $$(z_1, z_2, z_3, z_4) = \frac{(z_1 −z_3)(z_2 −z_4)}{(z_1 − z_4)(z_2 − z_3)};$$ if one of the points $z_k$ is $\infty$, cross out the two factors containing $z_k$.
Show that the cross ratio is $f(z_1)$, where $f$ is the unique linear fractional transformation sending $z_2 \mapsto 1$, $z_3 \mapsto 0$, and $z_4 \mapsto \infty$.
Given another four distinct points $w_1, w_2, w_3, w_4 \in \mathbb{C}\cup\{\infty\}$, show that there exists a linear fractional transformation sending $z_k \mapsto w_k$ for $k = 1, \dots, 4$ if and only if $(z_1, z_2, z_3, z_4) = (w_1, w_2, w_3, w_4)$.
Show that $(z_1, z_2, z_3, z_4)$ is real if and only if the four points $z_1, z_2, z_3, z_4$ lie on a line or a circle.