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Two Conditions for a Tetrahedron to Be Orthocentric

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An altitude of a tetrahedron is a line from a vertex perpendicular to the face opposite that vertex. A tetrahedron is orthocentric if the four altitudes meet at the same point, which is called the orthocenter or the Monge point.

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Let the opposite side lengths of a tetrahedron be and , and and and . Then is orthocentric if and only if .

A bimedian of a tetrahedron is a line segment that joins the midpoints of a pair of opposite edges. A tetrahedron is orthocentric if and only if the three bimedians have equal length.

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

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The proof can be found in [1, p. 123].

Reference

[1] V. V. Prasolov and I. F. Sharygin, Problems in Stereometry (in Russian), Moscow: Nauka, 1989.

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