In geometry, the **incircle** or **inscribed circle** of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.^{[1]}

An **excircle** or **escribed circle**^{[2]} of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.^{[3]}

The center of the incircle, called the **incenter**, can be found as the intersection of the three internal angle bisectors.^{[3]}^{[4]} The center of an excircle is the intersection of the internal bisector of one angle (at vertex $A$, for example) and the external bisectors of the other two. The center of this excircle is called the **excenter** relative to the vertex $A$, or the **excenter** of $A$.^{[3]} Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.^{[5]}^{:p. 182}

All regular polygons have incircles tangent to all sides, but not all polygons do; those that do are tangential polygons. See also Tangent lines to circles.

## Incircle and incenter

Suppose $\triangle ABC$ has an incircle with radius $r$ and center $I$.
Let $An$**excircle** or **escribed circle**^{[2]} of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.^{[3]}

The center of the incircle, called the **incenter**, can be found as the intersection of the three internal angle bisectors.^{[3]}^{[4]} The center of an excircle is the intersection of the internal bisector of one angle (at vertex $A$, for example) and the external bisectors of the other two. The center of this excircle is called the **excenter** relative to the vertex $A$, or the **excenter** of $A$.^{[3]} Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.^{[5]}^{:p. 182}

All regular polygons have incircles tangent to all sides, but not all polygons do; those that do are tangential polygons. See also Tangent lines to circles.

Suppose $\triangle ABC$ has an incircle with radius $r$ and center $I$.
Let $a$ be the length of $BC$, $b$ the length of $AC$, and $c$ the length of $AB$.
Also let $T_{A}$, $T_{B}$, and $T_{C}$ be the touchpoints where the incircle touches $BC$, $AC$, and $AB$.

### Incenter

The incenter is the point where the internal angle bisectors of $\angle ABC,\angle BCA,{\text{ and }}\angle BAC$ meet.

The distance from vertex $A$ to the incenter $I$ is:^{[citation needed]}

- $d(A,I)=c{\frac {\sin \left({\frac {B}{2}}\right)}{\cos \left({\frac {C}{2}}\right)}}=b{\frac {\sin \left({\f$
The incenter is the point where the internal angle bisectors of $\angle ABC,\angle BCA,{\text{ and }}\angle BAC$ meet.

The distance from vertex $}$

The distance from vertex $A$ to the incenter $I$ is:^{[citation needed]}

The trilinear coordinates for a point in the triangle is the ratio of all the distances to the triangle sides. Because the incenter is the same distance from all sides of the triangle, the trilinear coordinates for the incenter are^{[6]}

- $\ 1:1:1.$

#### Barycentric coordinates

The barycentric coordinates for a point in a triangle give weights such that the point is the weighted average of the triangle vertex positions.
Barycentric coordinates for the incenter are given by^{[citation needed]}

- $\ a:b:c$

where $a$, $$

The barycentric coordinates for a point in a triangle give weights such that the point is the weighted average of the triangle vertex positions.
Barycentric coordinates for the incenter are given by^{[citation needed]}

- $\ a:b:c$

where $a$$a$, $b$, and $c$ are the lengths of the sides of the triangle, or equivalently (using the law of sines) by

- $\mathrm{sin}(A):\mathrm{sin}(B):\mathrm{sin}(C)}where$$A$, $B$, and $C$ are the angles at the three vertices.
#### Cartesian coordinates

The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter (that is, using the barycentric coordinates given above, normalized to sum to unity) as weights. The weights are positive so the incenter lies inside the triangle as stated above. If the three vertices are located at $({x}_{}$$(x_{a},y_{a})$, $(x_{b},y_{b})$, and $(x_{c},y_{c})$, and the sides opposite these vertices have corresponding lengths $a$, $b$, and $c$, then the incenter is at^{[citation needed]}

- $$$r$ of the incircle in a triangle with sides of length $a$
*, $b$*, $c$ is given by^{[7]}
- $r={\sqrt {\frac {(s-a)(s-b)(s-c)}{s}}},$ where $s=(a+b+c)/2.$

See Heron's formula.

#### Distances to the vertices

Denoting the incenter of $\triangle ABC$ as $I$, the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation^{[8]}

- $\frac{IA\cdot IA}{CA\cdot AB}+\frac{IB\cdot IB}{}$
See Heron's formula.

#### Distances to the vertices

Denoting the incenter of $\triangle ABC$ as $I$, the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation^{[8]}

- $\triangle ABC$