---
tag: WRKFH25NCX
kind: concept
name: Triangle center
msc: [51M04]
sources:
  - url: https://en.wikipedia.org/wiki/Triangle_center
same_as:
  wikidata: Q778806
  wikipedia: Triangle center
---

## Summary

A triangle center is a point attached to every [[triangle|0CTNFGYMM2]] by a fixed rule.
The rule must not depend on the labelling of the vertices, and it must commute with similarities.
Classical examples are the [[centroid|0X7EEWES5G]], the [[incentre|0M9MX8YW0A]], the [[circumcentre|0F98FZK14A]] and the [[orthocentre|0WB619CA0V]].
The notion is made precise with trilinear coordinates, and Clark Kimberling's Encyclopedia of Triangle Centers catalogues many thousands of them.

## Explanations

### L1 human

Every triangle has several natural "middle points": the balance point, the centre of the inscribed circle, the centre of the circle through the corners, and so on.
A triangle center is any such point given by a fixed recipe.
The recipe must treat the three corners in the same way, so renaming the corners gives the same point.
It must also behave well under scaling: enlarge the triangle and the point moves with it.