I followed the link under that one factoring question which involved proving a+b+c+d is composite, and got to that IMO question discussion. At the bottom of the discussion I noticed Arun Iyer commenting on David Loeffler's publication of a Ptolemy-like theorem that encompasses all quadrilaterals, not just cyclic ones.
I don't have access to the Crux magazine, so I was wondering if you(David Loeffler), could post that theorem. Such a theorem would be very useful!

Let q = angle A B C+ angle A D C. Then

A C².B D²=A B².C D²+A D².B C²-2A B.B C.C D.D Acosq

(Evidently if A B C D is cyclic q = p, so this reduces to A C.B D=A B.C D+A D.B C, which is Ptolemy.)

I wasn't the first to discover this theorem. Apparently a German mathematician called Bretschneider got there first in the late 19th century. I haven't actually ever used it except for in the one problem that I was working on when I discovered it.

David

Even if you weren't the first I'd still say the theorem is pretty cool!