the joins of tjhe mid point s of the opposite edges of a tetrahedron intersect and bisect each other. proove it.

the joins of tjhe mid  point s of the opposite edges of a tetrahedron intersect and bisect each other.   proove it.


1 Answers

Aman Bansal
592 Points
12 years ago

Dear Prashant,

Every tetrahedron can be "inscribed" in a parallelepiped with volume three times the one of the tetrahedron.

proof: the computation is easy in the case of a regular tetrahedron inscribed in a cube with edge 1 (each of the four tri-right-angled tetrahedra has a volume of 1/6; thus it remains 1/3 for the regular tetrahedron).

Generalizing is easy: the four pyramids have the same volume (same height, that of the parallelepiped, and bases with same area, half of that of the parallelepiped s base).

The edges of the tetrahedron are six of the twelve diagonals of the parallelepiped s faces.

The six other diagonals of the parallelepiped s faces define a second tetrahedron with same volume. The union of these two tetrahedra is an "anti-parallelepiped" (distorted anticube).

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Aman Bansal

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