TITLE: Characterizations of the cross-ratio
QUESTION [5 upvotes]: $$
(z_1,z_2;z_3,z_4) = \frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)}
$$
What are the most prominent or most interesting theorems of the following form?
Theorem: The cross-ratio is the only function that satisfies this-particular-specified-condition.

REPLY [2 votes]: The cross-ratio is the unique rational function of four complex numbers up to multiplication by a constant such that the cross-ratio of four distinct points is real if and only if the points are concyclic.