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 $\triangle ABC$.

Then:

the excenter of $\triangle DEF$ with respect to $EF$ is $H$

the excenter of $\triangle DEF$ with respect to $DF$ is $B$

the excenter of $\triangle DEF$ with respect to $DE$ 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 $\triangle DEF$ with respect to $EF$ is $H$

the excenter of $\triangle DEF$ with respect to $DF$ is $B$

the excenter of $\triangle DEF$ with respect to $DE$ is $C$

and:

$A$ is the incenter of $\triangle DEF$

$\blacksquare$