This is a nice paper which finds a simple formula for the area of the medial parallelogram of a tetrahedron in terms of the lengths of the six edges. This is interesting to me because in simplicial models for quantum gravity, the formula is needed to deal the problem of length operators.

The paper finds that given a pair of non-incident edges in a tetrahedron, the medial parallelogram determined by the pair is the parallelogram whose vertices are the mid-points of the remaining four edges.

The area of the medial parallelogram determined by the edges of lengths d and e in the tetrahedron is

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