Loeffler Quadrilateral Theorem

By Anthony Cardell Tony on Wednesday, April 17, 2002 - 11:32 pm:

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!

By David Loeffler on Thursday, April 18, 2002 - 09:29 pm:

Err - in fact for this particular problem I think it's utterly useless, but here it is anyway:

Let $θ =$ angle $A B C +$ angle $A D C$ . Then

$A C^{2} . B D^{2} = A B^{2} . C D^{2} + A D^{2} . B C^{2} - 2 A B . B C . C D . D A \cos θ$

(Evidently if $A B C D$ is cyclic $θ = π$ , 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

By Anthony Cardell Tony on Friday, April 19, 2002 - 02:32 am:

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