Incircles

An incircle, or inscribed circle, is a circle that is tangent to all the sides of a polygon. In a triangle, every triangle has a unique incircle tangent to all three sides. The center of the incircle is called the incenter, the point where the triangle's internal angle bisectors meet, and the radius is the inradius.

In a triangle, the inradius r can be computed from the area Δ and semiperimeter s = (a+b+c)/2 by r = Δ / s. Since Δ = rs, this also gives r = 2Δ/(a+b+c). The incircle touches each side at a single point, and the incenter is equidistant from all three sides.

For a polygon with an incircle (a tangential polygon), a circle tangent to all its sides exists and is unique. Regular polygons are tangential; in such figures the common distance from the center to each side is the radius. The center is equidistant from the sides, and the touchpoints mark where the circle meets the sides.

Construction and related concepts: To construct the incircle of a triangle, draw the internal angle bisectors; their intersection is the incenter. A circle with radius equal to its distance to any side is the incircle. The circle’s points of tangency with each side are the touchpoints. Incircles have related notions such as excircles, which are tangent to one side and the extensions of the other sides.