Right-Angled Tetrahedron

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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.

Contributed by: Izidor Hafner (April 2017)
Open content licensed under CC BY-NC-SA


Snapshots


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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