The incenter I is the center of the incircle for a polygon or insphere for a polyhedron (when they exist). The corresponding radius of the incircle or insphere is known as the inradius.

The incenter can be constructed as the intersection of angle bisectors. It is also the interior point for which distances to the sides of the triangle are equal. It has trilinear coordinates 1:1:1, i.e., triangle center function


and homogeneous barycentric coordinates (a,b,c). It is Kimberling center X_1.

For a triangle with Cartesian vertices (x_1,y_1), (x_2,y_2), (x_3,y_3), the Cartesian coordinates of the incenter are given by


The distance between the incenter and circumcenter is sqrt(R(R-2r)), where R is the circumradius and r is the inradius, a result known as the Euler triangle formula.

The incenter lies on the Nagel line and Soddy line, and lies on the Euler line only for an isosceles triangle. The incenter is the center of the Adams' circle, Conway circle, and incircle. It lies on the Darboux cubic, M'Cay cubic, Neuberg cubic, orthocubic, and Thomson cubic. It also lies on the Feuerbach hyperbola.

For an equilateral triangle, the circumcenter O, triangle centroid G, nine-point center F, orthocenter H, and de Longchamps point Z all coincide with I.

The distances between the incenter and various named centers are given by


where F is the Feuerbach point, G is the triangle centroid, Ge is the Gergonne point, H is the orthocenter, K is the symmedian point, L is the de Longchamps point, M is the mittenpunkt, N is the nine-point center, Na is the Nagel point, Sp is the Spieker center, r is the inradius, R is the circumradius, Delta is the triangle area, and S_omega is Conway triangle notation.

The following table summarizes the incenters for named triangles that are Kimberling centers.

The incenter and excenters of a triangle are an orthocentric system.

The circle power of the incenter with respect to the circumcircle is


(Johnson 1929, p. 190).

If the incenters of the triangles DeltaA_1H_2H_3, DeltaA_2H_3A_1, and DeltaA_3H_1H_2 are X_1, X_2, and X_3, then X_2X_3 is equal and parallel to I_2I_3, where H_i are the feet of the altitudes and I_i are the incenters of the triangles. Furthermore, X_1, X_2, X_3, are the reflections of I with respect to the sides of the triangle DeltaI_1I_2I_3 (Johnson 1929, p. 193).

See also

Circumcenter, Cyclic Quadrilateral, Excenter, Gergonne Point, Incentral Triangle, Incircle, Inradius, Orthocenter, Orthocentric Centroid, Nagel Line

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Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p. 622, 1970.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 10, 1967.Dixon, R. Mathographics. New York: Dover, p. 58, 1991.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 182-194, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Incenter.", C. "Encyclopedia of Triangle Centers: X(1)=Incenter.", D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 115-116, 1991.

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Cite this as:

Weisstein, Eric W. "Incenter." From MathWorld--A Wolfram Web Resource.

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