Incircle and excircles of a triangle
excircles (blue), internal angle bisectors (red)
and external angle bisectors (green)
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.
contact triangle (red) and Gergonne point (green)
Denoting the three vertices of the triangle by A, B and C and the three points where the incircle touches the triangle by TA, TB and TC (where TA is opposite of A, etc.), the triangle TATBTC is known as the contact triangle of ABC. The incircle of ABC is the circumcircle of TATBTC. The three lines ATA, BTB and CTC 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