Monge Point of a Tetrahedron

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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.

[more]

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.

[less]

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


Snapshots


Details

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

Reference

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



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send