Quadrilateral in Complex Plane is Cyclic iff Cross Ratio of Vertices is Real

Theorem
Let $z_1, z_2, z_3, z_4$ be distinct complex numbers.

Then:
 * $z_1, z_2, z_3, z_4$ define the vertices of a cyclic quadrilateral

their cross ratio:
 * $\paren {z_1, z_3; z_2, z_4} = \dfrac {\paren {z_1 - z_2} \paren {z_3 - z_4} } {\paren {z_1 - z_4} \paren {z_3 - z_2} }$

is wholly real.