Tetrahedron centroid-to-midpoint sum of squares

Place the centroid of a tetrahedron at the origin, so its four vertex vectors satisfy $r_1+r_2+r_3+r_4=0$. Midpoints of opposite edges have opposite vectors, hence equal distances from the centroid. For the three opposite-edge pairs, choose midpoint vectors $(r_1+r_j)/2$, $j=2,3,4$. Expanding their squared lengths and using ${\sum }_{j=2}^4{r}_j=-r_1$ givesThe argument uses no regularity or equal-edge assumption on the tetrahedron.