The Area of the Medial Parallelogram of a Tetrahedron

The purpose of this note is to present a simple formula for the area of the medial parallelogram of a tetrahedron in terms of the lengths of the six edges, together with its derivation. It is not claimed that this formula is original, though diligent searches of standard sources of mensuration formulae, and inquiries to mathematicians in a variety of fields suggest that it is not generally known, and may be new. The author would be grateful for any references to previously published instances of this formula. Despite the very classical nature of the problem it solves, there is some serious contemporary interest arising from recently proposed simplicial models for quantum gravity (cf. [1], [2]), in which such a formula is needed to approach the problem of length operators. Consider a tetrahedron with edge-lengths as in Figure 1. Fix a pair of nonincident edges, say those of lengths d and e. It is then easy to see that the midpoints of the remaining four edges lie in a plane parallel to both of the chosen edges, and equidistant from the planes containing each chosen edge and parallel to both, and that they form the vertices of a parallelogram in this plane.