Incircle

Definition

Let $ABC$ be a triangle in the Euclidean plane $\mathbb{R}^2$, with incentre $I$ and inradius $r$. Let $d$ be the Euclidean distance. The incircle of $ABC$ is the circle 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 $T = r s$.