Excenters and Incenter of Orthic Triangle
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Theorem
Acute Triangle
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 $\triangle ABC$.
Obtuse Triangle
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 $A$.
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: The pedal triangle