= Solution
Translate one vertex of the <tetrahedron> to the origin and denote the other vertex vectors by $a,b,c$. Translation preserves edge vectors and their <dot products>. We may label the two known perpendicular pairs of <opposite edges of a tetrahedron> as the pairs represented by $a,c-b$ and $b,a-c$. Their orthogonality gives
$$
a\cdot c=a\cdot b,\qquad b\cdot a=b\cdot c.
$$
Hence all three pairwise <dot products> have the same value $s$. The third pair satisfies
$$
c\cdot(b-a)=c\cdot b-c\cdot a=s-s=0,
$$
so \b[the third pair of opposite edges is also perpendicular].
Expanding squared lengths using the <dot product> gives
$$
\begin{aligned}
|a|^2+|c-b|^2
&=|b|^2+|a-c|^2\\
&=|c|^2+|b-a|^2\\
&=|a|^2+|b|^2+|c|^2-2s.
\end{aligned}
$$
Thus \b[the sums of squared opposite-edge lengths are equal]. These two conclusions are the basic identities for <orthogonal opposite edges of a tetrahedron>.
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