---
tag: QM0CGSQ4TF
kind: concept
name: Fermat point
msc: [51M04, 51M16]
sources:
  - url: https://en.wikipedia.org/wiki/Fermat_point
same_as:
  wikidata: Q914929
  wikipedia: Fermat point
---

## Summary

The Fermat point of a [[triangle|0CTNFGYMM2]] is the point of the plane whose total distance to the three vertices is as small as possible.
When every angle of the triangle is less than $120^\circ$, it is the interior point that sees each side under $120^\circ$.
It is a model case of a geometric minimisation problem and of a [[triangle center|WRKFH25NCX]].

## Explanations

### L1 human

Three villages want to build one meeting point and join it to each village by a straight road.
The Fermat point is the place that makes the total length of road as short as possible.
At this point the three roads meet at equal angles of $120^\circ$, unless one corner of the triangle is already very blunt.
For an equilateral triangle the Fermat point is simply the centre.