Excircles

Excircles are circles associated with a triangle that are externally tangent to the triangle’s sides. For a triangle with side lengths a, b, c opposite vertices A, B, C and semiperimeter s = (a + b + c)/2, there are three excircles. Each excircle is tangent to one side of the triangle and to the extensions of the other two sides. The excircle opposite A is tangent to BC and to the lines AB and AC extended beyond B and C, respectively; analogously for the other two excircles.

The centers of the excircles are the excenters I_a, I_b, I_c. I_a is the intersection of the internal angle bisectors of B and C with the external angle bisector of A; I_b and I_c are defined similarly. All three excenters lie outside the triangle.

Let Δ be the area of the triangle. The inradius r = Δ/s. The radii of the excircles are r_a = Δ/(s − a), r_b = Δ/(s − b), and r_c = Δ/(s − c). These relationships follow from expressing the area as the product of a radius and the corresponding semiperimeter segment for each excircle.

In contrast to the incircle, which is tangent to all three sides, each excircle is tangent to exactly one side and to the extensions of the other two sides. The three excircles provide a complementary set of tangent circles associated with the triangle and are linked to the triangle’s angle bisectors and area through the exradii above.