---
tag: 93343FN5PF
kind: statement
role: theorem
name: Feuerbach's theorem
msc: [51M04]
uses: [0CTNFGYMM2, 033H8MQRC3, 07DFG5JYPT]
about: [033H8MQRC3, 07DFG5JYPT]
sources:
  - url: https://en.wikipedia.org/wiki/Feuerbach_point
---

## Statement

Let $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane that is not equilateral.
Then its [[nine-point circle|033H8MQRC3]] is internally tangent to its [[incircle|07DFG5JYPT]].
It is also externally tangent to each of its three excircles.
An excircle is a circle outside the triangle tangent to one side and to the extensions of the other two sides.

## Remarks

Let $R$ be the circumradius, $r$ the inradius, and $N$ and $I$ the centres of the nine-point circle and of the incircle.
The nine-point circle has radius $R/2$, and $N$ is the midpoint of the circumcentre $O$ and the [[orthocentre|0WB619CA0V]] $H$.
Write $a, b, c$ for the side lengths, and $A, B, C$ also for the angles.
Two known formulas give $OH^2 = 9R^2 - (a^2 + b^2 + c^2)$ and $IH^2 = 2r^2 - 4R^2 \cos A \cos B \cos C$.
With these and [[Euler's theorem|RFCQT6TKZF]] for $OI$, [[Apollonius's theorem|0NAKHQ9XYY]] in the triangle $OIH$ gives $IN = R/2 - r$.
This says the two circles are internally tangent; a similar computation gives $I_a N = R/2 + r_a$ for each excentre $I_a$ with exradius $r_a$.

In an equilateral triangle the nine-point circle and the incircle coincide, since $N = I$ and $R/2 = r$.
The point of tangency with the incircle is the Feuerbach point of the triangle.

## Explanations

### L1 human

Every triangle has a special circle through nine notable points, the nine-point circle.
It also has an inscribed circle and three circles that touch its sides from outside.
Feuerbach's theorem says the nine-point circle touches all four of these circles.
This is a surprising coincidence that holds for every triangle.

### L2 human

The theorem is a highlight of classical triangle geometry, published by Karl Wilhelm Feuerbach in 1822.
It ties together the [[circumcircle|03VZJE6C8C]], the incircle and the nine-point circle through the radii $R$, $r$ and $R/2$.