Excenters and Incenter of Orthic Triangle

Theorem

Let $\triangle ABC$ be an acute triangle.

Let $\triangle DEF$ be the orthic triangle of $\triangle ABC$ such that:

$D$ is on $BC$

$E$ is on $AC$

$F$ is on $AB$

Then:

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

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 the orthocenter of $\triangle ABC$.

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 $A$.

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: The pedal triangle