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

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 (n2). Extensions of the Möbius-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 (n3).