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
barycentric coordinates . It is Kimberling
triangle with Cartesian vertices , , , 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.
equilateral triangle, the circumcenter , triangle
nine-point center , orthocenter , and de Longchamps point all coincide with .
The distances between the incenter and various named centers are given by
is the Feuerbach
is the triangle centroid, is the Gergonne point,
is the orthocenter,
is the symmedian
is the de Longchamps point, is the mittenpunkt, is the nine-point center,
is the Nagel
is the Spieker center, is the inradius, is the circumradius, is the triangle
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.
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.
, , are the reflections of with respect to the sides of the triangle (Johnson 1929, p. 193).
See also Circumcenter
Explore with Wolfram|Alpha
References Carr, G. S. New York: Chelsea, p. 622,
1970. Formulas and Theorems in Pure Mathematics, 2nd ed. Coxeter, H. S. M. and Greitzer, S. L. Washington, DC: Math. Assoc. Amer., p. 10, 1967. Geometry
New York: Dover, p. 58, 1991. Mathographics. Johnson, R. A.
Boston, MA: Houghton Mifflin, pp. 182-194, 1929. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. 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." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X1. Wells,
D. London: Penguin,
pp. 115-116, 1991. The
Penguin Dictionary of Curious and Interesting Geometry. Referenced on Wolfram|Alpha Incenter
Cite this as:
Weisstein, Eric W. "Incenter." From --A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/Incenter.html