Isosceles tetrahedrons with integer edges and volume in the \(\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}\) grid (Q6082512)

An \(n\)-tetrahedron consists of four points in \({\mathbb R}^n\), with \(n\geq 2\), such that there are no three points on a line. After proving an identity involving the edge lengths of an \(n\)-tetrahedron and the diagonal lengths of its medial octahedron, it is shown how to construct, using Euler bricks (rectangular cuboids whose edges and face diagonals have integer lengths), in the grid \({\mathbb Z}\times{\mathbb Z}\times {\mathbb Z}\) a family of isosceles tetrahedrons with integer edge length and integer volume.