Let $z_1, z_2, z_3, z_4 ∈$ $\mathbb{C}$ such that these numbers are in quadrilateral appear in a cyclic and positive order, prove that these numbers are contained in a circumference if and only if:
$$(\frac{z_1 - z_2}{z_3 - z_2})(\frac{z_3 - z_4}{z_1 - z_4}) < 0$$
So, so far I have tried to prove this using geometry and tried some theorems of complex numbers, but since we only started to see complex numbers about one and a half week, I don't have a great knowledge about complex numbers.
By doing a little bit of research I find that this problem has to see a bit with Ptolemy's theorem. Am I rigth?
I also don't have seen Euler's, can you help me?