Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
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
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.