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The investigation of the relation among the distances of an arbitrary point in the Euclidean space $\mathbb{R}^n$ 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 [4], [10], [9], and [15], 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 $n \ge 3$, 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 [18]. 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 [9], [15], and [16]. Thus the paper is a further testimony to the special role that linear algebra plays in higher dimensional geometry.

If $S$ is a given regular $n$-simplex, $n \ge 2$, of edge length $a$, then the distances $a_1$, $\cdots$, $a_{n+1}$ of an arbitrary point in its affine hull to its vertices are related by the fairly known elegant relation $\phi_{n+1} (a,a_1,\cdots,a_{n+1})=0$, where $$\phi = \phi_t (x, x_1,\cdots,x_{n+1}) = \left( x^2+x_1^2+\cdots+x_{n+1}^2\right)^2 - t\left( x^4+x_1^4+\cdots+x_{n+1}^4\right).$$ The natural question whether this is essentially the only relation is answered positively by M. Hajja, M. Hayajneh, B. Nguyen, and Sh. Shaqaqha in a recently submitted paper entitled "Distances from the vertices of a regular simplex." In that paper, the authors made use of the irreducibility of the polynomial $\phi $ in the case when $n \ge 2$, $t=n+1$, $x= a \ne 0$, and $k = \mathbb{R}$, but supplied no proof, promising to do so in another paper that is turning out to be this one. It is thus the main aim of this paper to establish that irreducibility. In fact, we treat the irreducibility of $\phi$ without restrictions on $t$, $x$, $a$, and $k$. As a by-product, we obtain new proofs of results pertaining to the irreducibility of the general Cayley-Menger determinant that are more general than those established by C. D'Andrea and M. Sombra in Sib. J. Math. 46, 71--76.

If $S$ is a given regular $d$-simplex of edge length $a$ in the $d$-dimensional Euclidean space $\mathcal{E}$, then the distances $t_1$, $\cdots$, $t_{d+1}$ of an arbitrary point in $\mathcal{E}$ to the vertices of $S$ are related by the elegant relation $$(d+1)\left( a^4+t_1^4+\cdots+t_{d+1}^4\right)=\left( a^2+t_1^2+\cdots+t_{d+1}^2\right)^2.$$ The purpose of this paper is to prove that this is essentially the only relation that exists among $t_1,\cdots,t_{d+1}.$ The proof uses tools from analysis, algebra, and geometry.

We consider the group G of R-automorphisms of the polynomial ring R[x] especially in the case where R is the ring of integers modulo n. We describe conjugacy classes in G, and in the case where n = 4, we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G., Comment: 30 pages

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., Comment: 25 pages

We investigate the geometric properties of simplices in Euclidean d-dimensional space for which two or more of the analogues of the classical triangle centers (including the centroid, circumcenter, incenter, orthocenter or Monge point, and the Fermat-Torricelli point) coincide. We also investigate the geometric significance of the cevian line segments through a given center having the same length. We give a unified presentation, including known results for d=2 and d=3., Comment: 24 pages