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.
An incircle of a polygon is the two-dimensional case of an insphere of a solid.
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)
|
with an alternative trilinear equation given by
 |
(2)
|
(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
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
where 
is the semiperimeter,
so the inradius is
 |  |  |
(7)
|
 |  |  |
(8)
|
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).
