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

Abstract

By adding parallelogram identities, we prove an equation on the edge lengths of a tetrahedron and the diagonal lengths of its medial octahedron. By this and with the help of Euler bricks we find in the grid \mathbb{Z}\times\mathbb{Z}\times\mathbb{Z} a family of isosceles tetrahedrons with integer edge length and integer volume.