Excenters and Incenter of Orthic Triangle/Obtuse Triangle

Theorem
Let $\triangle ABC$ be an obtuse triangle such that $A$ is the obtuse angle.

Let $\triangle DEF$ be the orthic triangle of $\triangle ABC$ such that:
 * $D$ is on $BC$
 * $E$ is on $AC$ produced
 * $F$ is on $AB$ produced.

Let $H$ be the orthocenter of $\triangle ABC$.


 * Orthic-Triangle-Obtuse.png

Then:
 * the excenter of $\triangle DEF$ $EF$ is $H$
 * the excenter of $\triangle DEF$ $DF$ is $B$
 * the excenter of $\triangle DEF$ $DE$ is $C$

and:
 * the incenter of $\triangle DEF$ is $A$.

Proof
From Orthic Triangle of Obtuse Triangle:
 * $\triangle DEF$ is also the orthic triangle of $\triangle HBC$, which is an acute triangle.

It follows immediately from Excenters and Incenter of Orthic Triangle of Acute Triangle that:
 * the excenter of $\triangle DEF$ $EF$ is $H$
 * the excenter of $\triangle DEF$ $DF$ is $B$
 * the excenter of $\triangle DEF$ $DE$ is $C$

and:
 * $A$ is the incenter of $\triangle DEF$