A simplex is said to be orthocentric if its altitudes intersect in a common point, called its orthocenter. In this paper it is proved that if any two of the traditional centers of an orthocentric simplex (in any dimension) coincide, then the simplex is regular. Along the way orthocentric simplices in which all facets have the same circumradius are characterized, and the possible barycentric coordinates of the orthocenter are described precisely. In particular these barycentric coordinates are used to parametrize the shapes of orthocentric simplices. The substantial, but widespread, literature on orthocentric simplices is briefly surveyed in order to place the new results in their proper context, and some of the previously known results are given new proofs from the present perspective.Keywords: barycentric coordinates, centroid, circumcenter, equiareal simplex, equifacetal simplex, equiradial simplex, Gram matrix, incenter, Monge point, orthocenter, orthocentric simplex, rectangular simplex, regular simplex
IntroductionThis paper is a study of the geometric consequences of assumed coincidences of centers of a d-dimensional orthocentric simplex (or, simply, orthocentric d-simplex ) S in the d-dimensional Euclidean space, d ≥ 3, i.e., of a d-simplex S whose d + 1 altitudes have a common point H, called the orthocenter of S. The centers under discussion are the centroid G, the circumcenter C and the incenter I of S. For triangles, these centers are mentioned in Euclid's Elements, and in fact they are the only centers mentioned there. It is interesting that the triangle's orthocenter H, defined as the intersection of the three altitudes, is never mentioned in the Elements, and that nothing shows Euclid's awareness of the fact that the three altitudes are concurrent, see [23], p. 58. It is also worth mentioning that one of the most elegant proofs of that concurrence is due to C. F. Gauss, and A. Einstein is said to have prized this concurrence for its nontriviality and beauty. However, in contrast to the planar situation, the d + 1 altitudes of a d-simplex are not necessarily concurrent if d ≥ 3. We may think of this as a first manifestation of the reality that general d-simplices, d ≥ 3, do not have all the nice properties that triangles have.It is natural that, besides G, C, and I, we will also consider the orthocenter H of a d-simplex S regarding its coincidence with the other three centers. It is well-known that for d = 2 the coincidence of any two of the four mentioned centers yields a regular (or equilateral) triangle; see [37], page 78, and for triangle centers in general we refer to [5] and [32]. For d ≥ 3 this is no longer true, i.e., only weaker degrees of regularity are obtained, see [13] and [14] for recent results on this. One of these weaker degrees is the equifacetality of a
