Orthocentric Simplices and Biregularity

Abstract

It is well known that the incenter of a Euclidean triangle lies on its Euler line connecting the centroid and the circumcenter if and only if the triangle is isosceles. The analogue of this result in higher dimensions is studied for orthocentric simplices. It is shown that the incenter of an orthocentric simplex of any dimension lies on its Euler line if and only if this simplex can be expressed as the join of two regular simplices, with all edges connecting the two corresponding components having the same length. These are precisely the orthocentric simplices whose group of isometries has as fixed point set a line (or a point, in the special case of a regular simplex).