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.
An excircle or escribed circle 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.
The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex , for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex , or the excenter of . 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. but not all polygons do; those that do are tangential polygons. See also tangent lines to circles.
Incircle and incenter
Suppose has an incircle with radius and center . Let be the length of , the length of , and the length of . Also let , , and be the touchpoints where the incircle touches , , and .
Incenter
The incenter is the point where the internal angle bisectors of meet.
The distance from vertex to the incenter is:
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
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
where , , and are the lengths of the sides of the triangle, or equivalently (using the law of sines) by
where , , and 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 , , and , and the sides opposite these vertices have corresponding lengths , , and , then the incenter is at
Radius
The inradius of the incircle in a triangle with sides of length , , is given by
where is the semiperimeter.
The tangency points of the incircle divide the sides into segments of lengths and
See Heron's formula.
Distances to the vertices
Denoting the incenter of as , the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation
Additionally,
where and are the triangle's circumradius and inradius respectively.
Other properties
The collection of triangle centers may be given the structure of a group under coordinate-wise multiplication of trilinear coordinates; in this group, the incenter forms the identity element.
Incircle and its radius properties
Distances between vertex and nearest touchpoints
The distances from a vertex to the two nearest touchpoints are equal; for example:
Other properties
If the altitudes from sides of lengths , , and are , , and , then the inradius is one-third of the harmonic mean of these altitudes; that is,
The product of the incircle radius and the circumcircle radius of a triangle with sides , , and is
Some relations among the sides, incircle radius, and circumcircle radius are:
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle.
Denoting the center of the incircle of as , we have
and
The incircle radius is no greater than one-ninth the sum of the altitudes.
The squared distance from the incenter to the circumcenter is given by
and the distance from the incenter to the center of the nine point circle is
The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).