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.
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.
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.
Definition and constructionEdit
It is a theorem in Euclidean geometry that the three interior angle bisectors of a triangle meet in a single point. In Euclid's Elements, Proposition 4 of Book IV proves that this point is also the center of the inscribed circle of the triangle. The incircle itself may be constructed by dropping a perpendicular from the incenter to one of the sides of the triangle and drawing a circle with that segment as its radius.
The incenter lies at equal distances from the three line segments forming the sides of the triangle, and also from the three lines containing those segments. It is the only point equally distant from the line segments, but there are three more points equally distant from the lines, the excenters, which form the centers of the excircles of the given triangle. The incenter and excenters together form an orthocentric system.
The medial axis of a polygon is the set of points whose nearest neighbor on the polygon is not unique: these points are equidistant from two or more sides of the polygon. One method for computing medial axes is using the grassfire transform, in which one forms a continuous sequence of offset curves, each at some fixed distance from the polygon; the medial axis is traced out by the vertices of these curves. In the case of a triangle, the medial axis consists of three segments of the angle bisectors, connecting the vertices of the triangle to the incenter, which is the unique point on the innermost offset curve. The straight skeleton, defined in a similar way from a different type of offset curve, coincides with the medial axis for convex polygons and so also has its junction at the incenter.
Let the bisection of and meet at , and the bisection of and meet at , and and meet at .
A line that is an angle bisector is equidistant from both of its lines when measuring by the perpendicular. At the point where two bisectors intersect, this point is perpendicularly equidistant from the final angle's forming lines (because they are the same distance from this angles opposite edge), and therefore lies on its angle bisector line.
Relation to triangle sides and verticesEdit
The trilinear coordinates for a point in the triangle give the ratio of distances to the triangle sides. Trilinear coordinates
for the incenter are given by
The collection of triangle centers may be given the structure of a group under coordinatewise multiplication of trilinear coordinates; in this group, the incenter forms the identity element.
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.
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—i.e., 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
Distances to verticesEdit
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
The incenter must lie in the interior of a disk whose diameter connects the centroid G and the orthocenterH (the orthocentroidal disk), but it cannot coincide with the nine-point center, whose position is fixed 1/4 of the way along the diameter (closer to G). Any other point within the orthocentroidal disk is the incenter of a unique triangle.
The Euler line of a triangle is a line passing through its circumcenter, centroid, and orthocenter, among other points.
The incenter generally does not lie on the Euler line; it is on the Euler line only for isosceles triangles, for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers.
Denoting the distance from the incenter to the Euler line as d, the length of the longest median as v, the length of the longest side as u, the circumradius as R, the length of the Euler line segment from the orthocenter to the circumcenter as e, and the semiperimeter as s, the following inequalities hold:
Area and perimeter splittersEdit
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter; every line through the incenter that splits the area in half also splits the perimeter in half. There are either one, two, or three of these lines for any given triangle.
Relative distances from an angle bisectorEdit
Let X be a variable point on the internal angle bisector of A. Then X = I (the incenter) maximizes or minimizes the ratio along that angle bisector.
^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.
^ abcEncyclopedia of Triangle Centers Archived 2012-04-19 at the Wayback Machine, accessed 2014-10-28.
^Euclid's Elements, Book IV, Proposition 4: To inscribe a circle in a given triangle. David Joyce, Clark University, retrieved 2014-10-28.
^Johnson, R. A. (1929), Modern Geometry, Boston: Houghton Mifflin, p. 182.
^Blum, Harry (1967), "A transformation for extracting new descriptors of shape", in Wathen-Dunn, Weiant (ed.), Models for the Perception of Speech and Visual Form(PDF), Cambridge: MIT Press, pp. 362–380, In the triangle three corners start propagating and disappear at the center of the largest inscribed circle.
^Aichholzer, Oswin; Aurenhammer, Franz; Alberts, David; Gärtner, Bernd (1995), "A novel type of skeleton for polygons", Journal of Universal Computer Science, 1 (12): 752–761, doi:10.1007/978-3-642-80350-5_65, MR 1392429.
^Allaire, Patricia R.; Zhou, Junmin; Yao, Haishen (March 2012), "Proving a nineteenth century ellipse identity", Mathematical Gazette, 96: 161–165, doi:10.1017/S0025557200004277.
^Dragutin Svrtan and Darko Veljan, "Non-Euclidean versions of some classical triangle inequalities", Forum Geometricorum 12 (2012), 197–209. http://forumgeom.fau.edu/FG2012volume12/FG201217index.html
^Marie-Nicole Gras, "Distances between the circumcenter of the extouch triangle and the classical centers" Forum Geometricorum 14 (2014), 51-61. http://forumgeom.fau.edu/FG2014volume14/FG201405index.html
^Schattschneider, Doris; King, James (1997), Geometry Turned On: Dynamic Software in Learning, Teaching, and Research, The Mathematical Association of America, pp. 3–4, ISBN 978-0883850992
^Edmonds, Allan L.; Hajja, Mowaffaq; Martini, Horst (2008), "Orthocentric simplices and biregularity", Results in Mathematics, 52 (1–2): 41–50, doi:10.1007/s00025-008-0294-4, MR 2430410, S2CID 121434528, It is well known that the incenter of a Euclidean triangle lies on its Euler line connecting the centroid and the circumcenter if and only if the triangle is isosceles.
^Arie Bialostocki and Dora Bialostocki, "The incenter and an excenter as solutions to an extremal problem", Forum Geometricorum 11 (2011), 9-12. http://forumgeom.fau.edu/FG2011volume11/FG201102index.html
^Hajja, Mowaffaq, Extremal properties of the incentre and the excenters of a triangle", Mathematical Gazette 96, July 2012, 315-317.