A monster: Sergey Sadov's Ptolemy-like theorem

vess

The following Ptolemy-like criterion for the concyclicity of four points, in spite of its apparent simplicity, appears to have been only very recently discovered, and (no, I'm not joking) it is incredibly difficult to prove.

Theorem. The convex quadrangle $ABCD$ is inscribed in a circle if and only if
$AB \cdot BC \cdot CA + AC \cdot CD \cdot DA = BC \cdot CD \cdot DB + AB \cdot BD \cdot DA.$

--Vesselin

grobber

Hmm.. This definitely doesn't look right. Are you sure you weren't referring to this?

**Posted:** Wed Feb 09, 2005 12:21 pm

vess

Ah, yes, sorry. Blush

This has also been discussed here ... It is very strange since in this article (in Russian) has the following abstract: A convex quadrilateral with sides $a,b,c,d$ and diagonals $p,q$ is cyclic iff $abp-bcq+cdp-daq=0$ . This condition, in spite of its simplicity, appears to be unnoted and unexpectedly proof-resilient. We employ advanced methods of computer algebra and nonlinear analysis. (!!!) Confused

Isn't it funny? Very Happy

Haha, it's indeed trivial, after thinking a while! I didn't think on it initially, just posted it directly here after seeing it in the mentioned article Smile

. Check the article out and tell us what do you think! Wink

--Vesselin

**Posted:** Wed Feb 09, 2005 12:46 pm