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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.

Ataulfo
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Christie
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1 Answers1

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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.

Berci
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  • I know here is a theorem for opposite angles in a cyclic quadrilateral and I understand the straight line argument but how can I show that the sum cannot be a multiple of $\pi$ for any other shape? – Christie Mar 20 '18 at 23:01
  • Ahh.. I think I misunderstood your question. So, you get that $(z_1,z_2,z_3,z_4)$ real implies they on a line or circle. Its converse also holds: if $z_1,z_2,z_3,z_4$ are on a line or circle, then we get a real number for their cross ratio. – Berci Mar 20 '18 at 23:08
  • If the sum $\theta_1+\theta_2$ is $\pi$, then it's again an inscribed quadrangle.. – Berci Mar 21 '18 at 13:40