Date: May 28, 2013 1:59 PM
Author: Steve B
Subject: Whose theorem is this?

I posted this in the sci.math independent forum, but thought I ought to post it here as well.

Two special cases (involving altitudes and perpendicular bisectors) of the following are mentioned in Secrets of Triangles: A Mathematical Journey by Alfred S. Posamentier and Ingmar Lehmann, but as far as I can tell the general result is not mentioned. I was curious as to who first found it. It may simply be so obvious that its original discoverer is not known.

From P, a point in the interior of triangle ABC, drop perpendiculars to each of the sides of ABC. The feet of the perpendiculars (Fa, Fb, Fc) will divide BC into segments BFa and FaC of length a1 and a2, CA into segments CFb and FbA of length b1 and b2 and AB into segments AFc and FcB of lengths c1 and c2.

a1^2 + b1^2 + c1^2 = a2^2 + b2^2 + c2^2