Feuerbach's relation and Ptolemy's theorem in R n

Abstract

Ptolemy's equality for four points on a circle is related to a Feuerbach-type area relation. This suggested an extension of Ptolemy's inequality to a Feuerbach type volume relation between simplexes formed from n+2 points in Rn (n≥2). Extensions of the Mobius-Neuberg and Pompeiu Theorems in R2 are given for Rn. Ptolemy's inequality is also extended to convex n-gons in R2 yielding an extension of Fuhrmann's hexagon theorem to 2n-gons in R2 (n≥3).