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.
Permanent Citation