The Area of the Medial Parallelogram of a Tetrahedron

Abstract

When a pair of non-incident edges of a tetrahedron is chosen, the midpoints of the remaining 4 edges are the vertices of a planar parallelogram. A formula is given in terms of the six edge lengths for the area of this parallelogram. It is not claimed that this formula is new, but it is certainly not well-known. The author would be very grateful for a citation to an occurence of the formula in previously existing literature (by e-mail to [e-mail]). The result is of some current interest due to the work of Barbieri and Barrett/Crane on attempts to formulate simplicial versions of quantum gravity. The new version corrects the coefficient in the main result. A paper of the same title but with a vectorial version of the derivation of the area formula was published by the author in the American Mathematical Monthly 106 (10) December 1999 pp. 956-598.