Cayley–Menger determinant

In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a ${\textstyle n}$-dimensional simplex in terms of the squares of all of the distances between pairs of its vertices. The determinant is named after Arthur Cayley and Karl Menger.

The ${\displaystyle n(n-1)/2}$ pairwise distance polynomials between n points in a real Euclidean space are Euclidean invariants that are associated via the Cayley-Menger relations. These relations served multiple purposes such as generalising Heron's Formula, as well as computing the content of a n-dimensional simplex, and ultimately determining if any real symmetric matrix is a Euclidean distance matrix for some n + 1 points in the field of distance geometry.