exradii

Exradii are the radii of the excircles of a triangle. An excircle is a circle tangent to one side of the triangle and to the extensions of the other two sides. The center of the excircle opposite a given vertex is called the excenter for that vertex, formed by the intersection of the external angle bisector at that vertex with the internal angle bisectors at the other two vertices.

For a triangle with side lengths a, b, c, semiperimeter s = (a + b + c)/2, and area Δ, the radii of the excircles are

r_a = Δ/(s − a), r_b = Δ/(s − b), r_c = Δ/(s − c).

The inradius is r = Δ/s, and the area Δ can be found with Heron’s formula Δ = sqrt(s(s − a)(s − b)(s − c)). Equivalently, each exradius can be written as r_a = r · s/(s − a), and similarly for r_b and r_c.

Geometric interpretation and related facts: the three excenters are the vertices of the excentral triangle, and each excircle is tangent to the triangle’s side opposite its excenter as well as to the extensions of the other two sides. Exradii exist for all triangles, regardless of acute, right, or obtuse angles, and their values reflect how far the corresponding excircle must be drawn to touch the extended sides.

Example: in a 3-4-5 triangle labeled with a = 5, b = 4, c = 3, s = 6 and Δ = 6, the exradii are r_a = Δ/(s − a) = 6/(6 − 5) = 6, r_b = 6/(6 − 4) = 3, and r_c = 6/(6 − 3) = 2.