Incircle and Excircles of A Triangle - Other Excircle Properties

Other Excircle Properties

The distance d between the circumcenter and the center of an excircle is given by

where R is the circumradius and r_ex is the radius of the excircle.

The circular hull of the excircles is internally tangent to each of the excircles, and thus is an Apollonius circle. The radius of this Apollonius circle is where r is the incircle radius and s is the semiperimeter of the triangle.

The following relations hold among the inradius r, the circumradius R, the semiperimeter s, and the excircle radii r_'a, r_b, r_c:

The circle through the centers of the three excircles has radius 2R.

If H is the orthocenter of triangle ABC, then