= 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 $\sum_{j=2}^4r_j=-r_1$ gives
$$
\sum_{j=2}^4\left|\frac{r_1+r_j}{2}\right|^2=\frac14\sum_{i=1}^4|r_i|^2.
$$
The argument uses no regularity or equal-edge assumption on the tetrahedron.
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