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The incenter 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 . It is Kimberling center .

For a triangle with vertices in Cartesian coordinates , , , the Cartesian coordinates of the incenter are given by

The distance between the incenter and circumcenter is , where is the circumradius and 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 , triangle centroid , nine-point center , orthocenter , and de Longchamps point all coincide with .

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

where is the Feuerbach point, is the triangle centroid, is the Gergonne point, is the orthocenter, is the symmedian point, is the de Longchamps point, is the mittenpunkt, is the nine-point center, is the Nagel point, is the Spieker center, is the inradius, is the circumradius, is the triangle area, and 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 , , and are , , and , then is equal and parallel to , where are the feet of the altitudes and are the incenters of the triangles. Furthermore, , , , are the reflections of with respect to the sides of the triangle (Johnson 1929, p. 193).

Reflecting the incenter of a triangle in its side midpoints gives the outer García triangle. Reflecting it in the perpendicular bisectors of the sides gives the inner García triangle (García 2014).

See also

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

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References

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.García, E. A. J. "A Note on Reflections." Forum Geom. 14, 155-161, 2014. https://drive.google.com/file/d/1HgLcT5S9immBULeiMr1IqPHbSCY4OTdo/view.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." https://faculty.evansville.edu/ck6/tcenters/class/incenter.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(1)=Incenter." https://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X1.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, pp. 115-116, 1991.

Referenced on Wolfram|Alpha

Incenter

Cite this as:

Weisstein, Eric W. "Incenter." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Incenter.html

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