Monge Point of a Tetrahedron

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A Monge plane of a tetrahedron is a plane through the midpoint of an edge and perpendicular to the opposite edge. Monge proved that the six Monge planes intersect at a point, known as the Monge point.


This Demonstration constructs a tetrahedron with edges of selected lengths using a Cayley–Menger determinant, then constructs the Monge point (shown as a small red sphere).

The altitudes of a general tetrahedron usually do not meet, but if the tetrahedron is orthocentric, then the orthocenter coincides with the Monge point. The "orthocentric" checkbox changes the parameters to give an orthocentric tetrahedron.


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



The proof can be found in [1, p. 135].


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

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