Points, Lines & Planes · Superchallenge problem

A tetrahedron has six edges whose lengths are $4$, $5$, $6$, $7$, $8$, and $9$ in some order. Two edges are called opposite if they share no vertex, and the segment joining the midpoints of two opposite edges is called a bimedian; the tetrahedron has three bimedians. Find the sum of the squares of the lengths of the three bimedians.