Excenters and Incenter of Orthic Triangle/Obtuse Triangle
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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:
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 $\triangle DEF$
$\blacksquare$