---
tag: 0J768YF1ZE
kind: definition
name: Nine-point circle
msc: [51M04]
defines: 033H8MQRC3
uses: [0CTNFGYMM2, 0S706JBZBG, 03VZJE6C8C, 0C9MZY2RNA, 0WB619CA0V]
sources:
  - url: https://en.wikipedia.org/wiki/Nine-point_circle
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Nine-point_circle
    locator: Discovery
  - url: https://en.wikipedia.org/wiki/Nine-point_circle
    locator: Other properties of the nine-point circle
---

## Definition

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane $\mathbb{R}^2$. Let $M_A$, $M_B$ and $M_C$ be the [[midpoints|0S706JBZBG]] of the sides $BC$, $CA$ and $AB$.
These three points are not collinear. The nine-point circle of $ABC$ is the [[circumcircle|03VZJE6C8C]] of the triangle $M_A M_B M_C$.

## Remarks

The circle also contains the feet of the three [[altitudes|0C9MZY2RNA]], and the midpoints of the segments $AH$, $BH$ and $CH$, where $H$ is the [[orthocentre|0WB619CA0V]]. This gives the nine points of the name.
Its centre is the midpoint of $H$ and the circumcentre $O$ of $ABC$. Its radius is half of the circumradius of $ABC$.
Brianchon and Poncelet proved in 1821 that all nine points lie on one circle. In 1822, Feuerbach proved that the circle touches the incircle and the three excircles.

## Explanations

### L1 human

Join the midpoints of the sides of a triangle to get a smaller triangle. The circle through its corners is the nine-point circle. It also passes through six other special points of the big triangle.