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.*]{} ]{}

[**Keywords:**]{} Cayley-Menger determinant; circumscriptible simplex; discriminant; homogeneous polynomial; irreducible polynomial; isodynamic simplex; isogonic simplex; orthocentric simplex; Pompeiu’s theorem; pre-kites; quadratic polynomial; regular simplex; symmetric polynomial; tetra-isogonic simplex; volume of a simplex

Introduction {#111}
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Let $S=[A_1, \cdots, A_{n+1}]$, $n \ge 2$, be a regular $n$-simplex of edge length $a$, and let $B$ be an arbitrary point in its affine hull. It is fairly well known that the distances $a_1,\cdots,a_{n+1}$ from $B$ to the vertices of $S$ satisfy the elegant relation $$\begin{aligned}
 \label{f-a}
\left(a^2+a_1^2+\cdots+a_{n+1}^2\right)^2 &=& (n+1)\left(a^4+a_1^4+\cdots+a_{n+1}^4\right);\end{aligned}$$ see [@Bentin-1]. The natural question whether this is (essentially) the only relation was posed and answered in the affirmative in [@Shadi]. However, the proof rests heavily on a result whose proof was not included therein, but was postponed for another paper, namely this one. This result states that the polynomial obtained from (\[f-a\]) by thinking of $a_1,\cdots,a_{n+1}$ 