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

(1)

and homogeneous barycentric coordinates . It is Kimberling
center .

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

(2)

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

(15)

(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).

See also Circumcenter ,

Cyclic Quadrilateral ,

Excenter ,

Gergonne
Point ,

Incentral Triangle ,

Incircle ,

Inradius ,

Orthocenter ,

Orthocentric Centroid ,

Nagel
Line
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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. 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."
http://faculty.evansville.edu/ck6/tcenters/class/incenter.html . Kimberling,
C. "Encyclopedia of Triangle Centers: X(1)=Incenter." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X1 . Wells,
D. The
Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 115-116, 1991. Referenced on Wolfram|Alpha Incenter
Cite this as:
Weisstein, Eric W. "Incenter." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Incenter.html

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