Excircle

An excircle of a triangle is a circle that is tangent to one side of the triangle and to the extensions of the other two sides. Every triangle has three excircles, one opposite each vertex.

Each excircle is centered at an excenter. The excenter opposite A is the intersection of the internal angle bisector of angle A with the external angle bisectors of angles B and C. The other two excenters are defined analogously for the opposite vertices.

An excircle opposite A is tangent to the side BC and tangent to the lines AB and AC extended beyond B and C. The corresponding tangency points lie on BC and on the extensions of AB and AC.

Let a = BC, b = CA, c = AB, s = (a + b + c)/2 and Δ be the area of the triangle. The exradius opposite A is r_a = Δ/(s − a); similarly r_b = Δ/(s − b) and r_c = Δ/(s − c). The incircle has radius r = Δ/s. Distances from the excenters to the sides follow these relations.

The excenters form the excentral triangle I_a I_b I_c. The original triangle’s incenter I is the orthocenter of this excentral triangle, and each excenter lies outside the original triangle. Excircles play a key role in various triangle-geometry constructions and identity proofs, complementing the properties of the incircle.