  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.
