Triangle is Orthic Triangle of Triangle formed from Excenters

Theorem
Let $\triangle ABC$ be a triangle whose sides are $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.

Let $I$ be the incenter of $\triangle ABC$.

Let $I_a$, $I_b$ and $I_c$ be the excenters of $\triangle ABC$ $a$, $b$ and $c$ respectively.

Let $\triangle I_a I_b I_c$ be the triangle whose vertices are those excenters of $I_a$, $I_b$ and $I_c$.

Then:
 * $\triangle ABC$ is the orthic triangle of $\triangle I_a I_b I_c$

and:
 * $I$ is the orthocenter of $\triangle I_a I_b I_c$.

Proof

 * Orthic-Triangle-of-Excenters.png

From Construction of Excircle to Triangle, it is seen that:
 * $A I_b$ is the angle bisector of $\angle PAC$
 * $A I_c$ is the angle bisector of $\angle QAB$.

Hence $I_b A I_c$ is a straight line.

From the construction in Excenters and Incenter of Orthic Triangle, $A I I_a$ is a straight line which is perpendicular to $I_b A I_c$.

The same argument is used to show that:
 * $B I I_b$ is a straight line which is perpendicular to the straight line $I_a B I_c$

and
 * $C I I_c$ is a straight line which is perpendicular to the straight line $I_a C I_b$.

It follows by definition that :
 * $\triangle ABC$ is the orthic triangle of $\triangle I_a I_b I_c$

and:
 * $I$ is the orthocenter of $\triangle I_a I_b I_c$.