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Something I've been wondering.

Suppose $z_1,z_2,z_3,z_4$ are consecutive vertices of a quadrilateral that lie in a circle in the complex plane.

Why does $$ |z_1-z_3||z_2-z_4|=|z_1-z_2||z_3-z_4|+|z_2-z_3||z_1-z_4|? $$ It's clear the equality $$ (z_1-z_3)(z_2-z_4)=(z_1-z_2)(z_3-z_4)+(z_2-z_3)(z_1-z_4) $$ holds just by elementary algebra. Dividing gives $$ \frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)}=\frac{(z_1-z_2)(z_3-z_4)}{(z_2-z_3)(z_1-z_4) }+1. $$ So the desired equality follows if both those quotients are positive. I also observe that the first quotient is just the cross ratio $(z_1,z_2,z_3,z_4)$ and the second quotient is just $-(z_1,z_3,z_2,z_4)$. Is there some extra information I'm not seeing that shows that these two quantities are both positive? Thank you.

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Talking about the quotients being positive does not make much sense, unless you can prove they are real numbers. – Aryabhata Feb 8 '12 at 22:09

This is just a restatement of Ptolemy's theorem.

You can find a complex number proof on the wiki page above and using that you should be able to complete your proof attempt.

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