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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 ${\displaystyle A}$, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex ${\displaystyle A}$, or the excenter of ${\displaystyle A}$.[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 ${\displaystyle \triangle ABC}$ has an incircle with radius ${\displaystyle r}$ and center ${\displaystyle I}$. Let

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 ${\displaystyle A}$, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex ${\displaystyle A}$, or the excenter of ${\displaystyle A}$.[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 ${\displaystyle \triangle ABC}$ has an incircle with radius ${\displaystyle r}$ and center ${\displaystyle I}$. Let ${\displaystyle a}$ be the length of ${\displaystyle BC}$, ${\displaystyle b}$ the length of ${\displaystyle AC}$, and ${\displaystyle c}$ the length of ${\displaystyle AB}$. Also let ${\displaystyle T_{A}}$, ${\displaystyle T_{B}}$, and ${\displaystyle T_{C}}$ be the touchpoints where the incircle touches ${\displaystyle BC}$, ${\displaystyle AC}$, and ${\displaystyle AB}$.

### Incenter

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

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

${\displaystyle d(A,I)=c{\frac {\sin \left({\frac {B}{2}}\right)}{\cos \left({\frac {C}{2}}\right)}}=b{\frac {\sin \left({\f$

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

The distance from vertex

The distance from vertex ${\displaystyle A}$ to the incenter ${\displaystyle I}$ 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]

${\displaystyle \ 1:1:1.}$

#### 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]

${\displaystyle \ a:b:c}$

where ${\displaystyle a}$,

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]

${\displaystyle \ a:b:c}$

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