11284

Right-Angled Tetrahedron

Let be a tetrahedron with the three plane angles at all right angles, that is, . (This is more explicitly known as a trirectangular tetrahedron.) Let , , . Then . The lines that join the midpoints of opposite edges are equal and meet at a point. The proof, outlined in the Details, implies that these three lines are diagonals of a rectangular prism, intersecting at the center.

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DETAILS

Proof
Let . Then , , , . The length of can be evaluated from triangle using the Pythagorean theorem, , and from triangle using the law of cosines, , giving
.
Simplifying this identity, we find: [1, pp. 102 and 117].
Reference
[1] V. V. Prasolov and I. F. Sharygin, Problems in Stereometry (in Russian), Moscow: Nauka, 1989.
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