Incircle and excircles of a triangle

A triangle (black) with incircle (purple), excircles (blue), internal angle bisectors (red) and external angle bisectors (green)

In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.

The center of the incircle can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle and the external bisectors of the other two. From this, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.

The radii of the in- and excircles are closely related to the area of the triangle. If S is the triangle's area and its sides are a, b and c, then the radius of the incircle (also known as the inradius) is S/(2(a+b+c)), the excircle at side a has radius S/(2(-a+b+c)), the excircle at side b has radius S/(2(a-b+c)) and the excircle at side c has radius S/(2(a+b-c)). From these formulas we see in particular that the excircles are always larger than the incircle, and that the largest excircle is the one attached to the longest side.

A triangle with incircle (black), contact triangle (red) and Gergonne point (green)

The triangle's nine point circle is tangent to the three excircles as well as to the incircle. The triangle's Feuerbach point lies on the incircle.

Denoting the three vertices of the triangle by A, B and C and the three points where the incircle touches the triangle by T_A, T_B and T_C (where T_A is opposite of A, etc.), the triangle T_AT_BT_C is known as the contact triangle of ABC. The incircle of ABC is the circumcircle of T_AT_BT_C. The three lines AT_A, BT_B and CT_C intersect in a single point, the triangle's Gergonne point G.

The Gergonne point of a triangle is equal to the symmedian point of its contact triangle.

See also: circumcircle