The point of intersection of angle bisectors of the 3 angles of triangle ABC is the incenter (denoted by I). The incircle (whose center is I) touches each side of the triangle.
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.
Together with the centroid, circumcenter, and orthocenter, it is one of the four triangle centers known to the ancient Greeks, and the only one of the four that does not in general lie on the Euler line. It is the first listed center, X(1), in Clark Kimberling's Encyclopedia of Triangle Centers, and the identity element of the multiplicative group of triangle centers.[1][2]
For polygons with more than three sides, the incenter only exists for tangential polygons - those that have an incircle that is tangent to each side of the polygon. In this case the incenter is the center of this circle and is equally distant from all sides.
^Kimberling, Clark (1994), "Central Points and Central Lines in the Plane of a Triangle", Mathematics Magazine, 67 (3): 163–187, doi:10.1080/0025570X.1994.11996210, JSTOR 2690608, MR 1573021.
^Encyclopedia of Triangle Centers Archived 2012-04-19 at the Wayback Machine, accessed 2014-10-28.
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement...
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incenter, centroid, or orthocenter coincide.: p.37 It is also equilateral if its circumcenter coincides with the Nagel point, or if its incenter coincides...
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in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained...
Angle bisectors (concur at incenter I) Lines from each vertex to P Lines to P reflected about the angle bisectors (concur at P*, the isogonal conjugate...
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the triangle and bisecting the associated angle. They all meet at the incenter. Medians connect each vertex of a triangle to the midpoint of the opposite...
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