Various kinds of centres of simplices

Abstract

A triangle in the Euclidean plane has various kinds of centres. For example the centroid G, the circumcentre O, the incentre I, the orthocentre H, and the cleavance centre J. In this paper, we find higher dimensional analogous of these centres for simplices in Euclidean n-space and also in spherical n-space. Each centre is described as the point of intersection of certain hyperplanes (or great hyperspheres in the spherical case). Several of the theorems relating the various kinds of centres for triangles are generalized to higher dimensions, for example we show that the centres O, G and H are colinear and similarly, that the centres J, G and I are colinear for any simplex in Euclidean n-space. Finally, we obtain some new characterizations for several of these centres.