---
tag: 0RGWYWWBM5
kind: definition
name: Incircle
msc: [51M04]
defines: 07DFG5JYPT
uses: [0M9MX8YW0A, 060W5CBZP6]
sources:
  - url: https://en.wikipedia.org/wiki/Incircle_and_excircles
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Incircle_and_excircles
    locator: Relation to area of the triangle
---

## Definition

Let $ABC$ be a triangle in the Euclidean plane $\mathbb{R}^2$, with [[incentre|0M9MX8YW0A]] $I$ and inradius $r$. Let $d$ be the Euclidean distance.
The incircle of $ABC$ is the [[circle|060W5CBZP6]] with the centre $I$ and the radius $r$:
$$\{\, X \in \mathbb{R}^2 \;:\; d(X, I) = r \,\}.$$

## Remarks

The incircle touches each side of the triangle at exactly one point, the foot of the perpendicular from $I$ to that side.
Let $T$ be the area of $ABC$ and $s$ half of its perimeter. Then $T = r s$, so $r = T / s$.
An excircle is a circle outside the triangle that touches one side and the extensions of the other two sides. Each triangle has three excircles.

## Explanations

### L1 human

The incircle is the biggest circle that fits inside a triangle. It touches each side once. Cut the triangle into three pieces from $I$ to the corners: each piece has the height $r$, and this gives $T = r s$.