  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.
