Prove that the cross ratio of four distinct points is real iff the four points lie on single Euclidean line or circle

Question

I have started this proof by rewriting the formula for the cross ratio in terms of the polar decomposition of complex numbers:

$r=\Big(\dfrac{z_1-z_3}{z_1-z_4}\Big)\Big(\dfrac{z_2-z_4}{z_2-z_3}\Big)=\Big|\dfrac{z_1-z_3}{z_1-z_4}\Big|\Big|\dfrac{z_2-z_4}{z_2- z_3}\Big|e^{i(\theta_1+\theta_2)}$

So now I know that for this to be real I need $\theta_1+\theta_2$ to be a multiple of $\pi$ but how can I prove that this only holds for circles or Euclidean lines?

I know there are other ways to prove this but this is the proof that follows the flow of the project I am working on so would like to try and continue with this proof.

Answer 1

Hint: We either have $\theta_1\equiv \theta_2\equiv 0\pmod{\pi} $ when they lie on a line.
In any other cases you can use a theorem for inscribed quadrangles to conclude they lie on a circle.