Incircle

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).