Question

Show that the joins of the mid-points of the opposite edges of a terahedron intersect and bisect each other.

Answer 1

$A,B,C,D$ are one pair of opposite edges.
[ The other pair are $BC$ and $AD$; $CA$ and $BD$]
the mid points $AB$ and $CD$ are $\frac{\overline{a}+\overline{b}}{2};\frac{\overline{c}+\overline{d}}{2}$
The mid point of line joining these two points is $\frac{\frac{\overline{a}+\overline{b}}{2}+\frac{\overline{c}+\overline{d}}{2}}{2}=\frac{\overline{a}+\overline{b}+\overline{c}+\overline{d}}{4}$
Similarly, we can prove that the line joining the midpoints of $BC$ and $AD$ $(R,S)$
also pass through this point. Hence, they bisect each other.