Pre-kites: simplices having a regular facet

The investigation of the relation among the distances of an arbitrary point in the Euclidean space Rn to the vertices of a regular n-simplex in that space has led us to the study of simplices having a regular facet. Calling an n-simplex with a regular facet an n-pre-kite, we investigate, in the spirit of Hajja and Walker (Int J Math Ed Sci Tech 32:501–508, 2001), Edmonds et al. (Beitr Algebra Geom 46:491–512, 2005b), Edmonds et al. (Results Math 47:266–295, 2005a), and Hajja (Results Math 49:237–263, 2006), and using tools from linear algebra, the degree of regularity implied by the coincidence of any two of the classical centers of such simplices. We also prove that if n≥ 3 , then the intersection of the family of n-pre-kites with any of the four known special families is the family of n-kites, thus extending the result in Hajja et al. (Beitr Algebra Geom 56:269–277, 2015). A basic tool is a closed form of a determinant that arises in the context of a certain Cayley–Menger determinant, and that generalizes several determinants that appear in Edmonds et al. (Results Math 47:266–295, 2005a), Hajja (Results Math 49:237–263, 2006), and Hajja and Hayajneh (J Geom 105:539–560, 2014). Thus the paper is a further testimony to the special role that linear algebra plays in higher dimensional geometry. © 2017, The Managing Editors.