Excenters and Incenter of Orthic Triangle/Acute 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 $\triangle ABC$.
Proof
From Altitudes of Triangle Bisect Angles of Orthic Triangle, $AD$ is the angle bisector of $\angle FDE$.
From Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular, the angle bisector of $\angle PDE$ is perpendicular to $AD$.
The line perpendicular to $AD$ is $BC$.
Similarly, from Altitudes of Triangle Bisect Angles of Orthic Triangle, $BE$ is the angle bisector of $\angle FED$.
From Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular, the angle bisector of $\angle DEQ$ is perpendicular to $BE$.
The line perpendicular to $BE$ is $AC$.
From Construction of Excircle to Triangle, the intersection of $AC$ and $BC$ is the excenter of $\triangle DEF$ with respect to $DE$.
The same argument can be used mutatis mutandis to demonstrate the locations of the excenter of $\triangle DEF$ with respect to $DF$ and $EF$.
$\Box$
From Altitudes of Triangle Bisect Angles of Orthic Triangle:
- $AD$ is the angle bisector of $\angle FDE$
- $BE$ is the angle bisector of $\angle DEF$
- $FC$ is the angle bisector of $\angle EFD$
From Line from Vertex of Triangle to Incenter is Angle Bisector it follows that $H$ is the incenter of $\triangle DEF$.
$\blacksquare$