# Equality on absolute values in the complex plane.

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