Distance between Incenter and Excenter of Triangle in Terms of Circumradius/Proof

Proof

 * Orthic-Triangle-of-Excenters.png

From Triangle is Orthic Triangle of Triangle formed from Excenters, we establish that $\triangle ABC$ is the orthic triangle of $\triangle I_a I_b I_c$.

By the Nine Point Circle Theorem, the Feuerbach circle of $\triangle I_a I_b I_c$ passes through each of $A$, $B$ and $C$.

Therefore the Feuerbach circle of $\triangle I_a I_b I_c$ is the circumcircle of $\triangle ABC$.

Hence the radius of the Feuerbach circle is $R$.

From Radius of Circumcircle is Twice Radius of Feuerbach Circle, the radius of the circumcircle of $\triangle I_a I_b I_c$ is $2 R$.

Hence: