---
tag: NW57FTPF3E
kind: concept
name: Monge point
msc: [51M04, 51M20]
sources:
  - url: https://en.wikipedia.org/wiki/Tetrahedron
---

## Summary

The Monge point of a [[tetrahedron|JFX8ASQS3E]] is the common point of six planes.
Each of them passes through the midpoint of an edge and is perpendicular to the opposite edge.
It is the reflection of the circumcentre in the [[centroid|0X7EEWES5G]].
It plays the role in a tetrahedron that the [[orthocentre|0WB619CA0V]] plays in a triangle, because the altitudes of a tetrahedron need not meet.
The definition extends to every [[simplex|YX1DJK2QAG]] of dimension at least $2$, and for a triangle it gives the orthocentre.

## Explanations

### L1 human

In a triangle, the three altitudes always meet at one point, the orthocentre.
In a tetrahedron, the four altitudes usually miss each other.
Gaspard Monge found a replacement: for each edge, take the plane through its midpoint at right angles to the opposite edge.
The six planes always meet at one point, the Monge point.
It lies on the line through the circumcentre and the centroid, on the other side of the centroid at the same distance.