In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be 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] 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 ${\displaystyle a}$ be the length of ${\displaystyle {\overline {BC}}}$, ${\displaystyle b}$ the length of ${\displaystyle {\overline {AC}}}$, and ${\displaystyle c}$ the length of ${\displaystyle {\overline {AB}}}$. Also let ${\displaystyle T_{A}}$, ${\displaystyle T_{B}}$, and ${\displaystyle T_{C}}$ be the touchpoints where the incircle touches ${\displaystyle {\overline {BC}}}$, ${\displaystyle {\overline {AC}}}$, and ${\displaystyle {\overline {AB}}}$.

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]}

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

Trilinear coordinates

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

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

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

$${\displaystyle \sin A:\sin B:\sin C}$$

where ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ 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 ${\displaystyle (x_a,y_{}}$Zdroj:https://en.wikipedia.org?pojem=Intouch_triangle
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