An incircle is an inscribed circle of a polygon, i.e., a circle that is tangent to each
of the polygon's sides. The center  of the incircle
is called the incenter, and the radius  of the circle is
called the inradius.
While an incircle does not necessarily exist for arbitrary polygons, it exists and is moreover unique for triangles,
regular polygons, and some other
polygons including rhombi, bicentric polygons, and tangential quadrilaterals.
The incenter is the point of concurrence of the triangle's angle bisectors. In addition, the points  ,  , and  of intersection
of the incircle with the sides of  are the
polygon vertices of the pedal triangle taking the incenter
as the pedal point (c.f. tangential triangle). This triangle
is called the contact triangle.
The trilinear coordinates of the incenter of a triangle are
 .
The polar triangle of the incircle
is the contact triangle.
The incircle is tangent to the
nine-point circle.
Pedoe (1995, p. xiv) gives a geometric
construction for the incircle.
There are four circles that are tangent to all three sides (or their extensions) of a given triangle:
the incircle  and three excircles  ,  , and  . These four
circles are, in turn, all touched by the nine-point
circle  .
The circle function of the incircle
is given by
 |
(1)
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with an alternative trilinear equation given by
 |
(2)
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(Kimberling 1998, p. 40).
The incircle is the radical circle of the tangent circles centered
at the reference triangle
vertices.
Kimberling centers  lie on the incircle for  (Feuerbach point), 1317, 1354, 1355, 1356, 1357, 1358, 1359,
1360, 1361, 1362, 1363, 1364, 1365, 1366, 1367, 2446, 2447, 3023, 3024, and 3025.
The area  of the triangle  is given
by
where  is the semiperimeter,
so the inradius is
Using the incircle of a triangle as the inversion center, the sides
of the triangle and its circumcircle are carried into four equal circles (Honsberger 1976, p. 21).
Let a triangle  have an incircle with incenter  and let the incircle
be tangent to  at  ,  , (and  ; not shown).
Then the lines  ,  , and the
perpendicular to  through  concur in a point  (Honsberger 1995).
Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing
an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl.
Dublin: Hodges, Figgis, & Co., pp. 53-55, 1888.
Coxeter, H. S. M. and Greitzer, S. L. "The Incircle and Excircles." §1.4 in Geometry Revisited. Washington, DC: Math. Assoc. Amer.,
pp. 10-13, 1967.
Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer.,
1976.
Honsberger, R. "An Unlikely Concurrence." §3.4 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry.
Washington, DC: Math. Assoc. Amer., pp. 31-32, 1995.
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. "Triangle Centers and Central Triangles." Congr. Numer. 129,
1-295, 1998.
Lachlan, R. "The Inscribed and the Escribed Circles." §126-128 in An Elementary Treatise on Modern Pure Geometry. London:
Macmillian, pp. 72-74, 1893.
Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC:
Math. Assoc. Amer., 1995.
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