Altitudes of Triangle Bisect Angles of Orthic Triangle

Theorem
Let $\triangle ABC$ be a triangle.

Let $\triangle DEF$ be its orthic triangle.

The altitudes of $\triangle ABC$ are the angle bisectors of $\triangle DEF$.

Proof

 * Pedal-Triangle-Angle-Bisectors.png

Consider the triangles $\triangle ABE$ and $\triangle ACF$.

We have that:
 * $\angle FAC$ and $\angle BAE$ are common
 * $\angle AFC$ and $\angle AEB$ are both right angles

and it follows from Triangles with Two Equal Angles are Similar that $\triangle ABE$ and $\triangle ACF$ are similar.

Thus:
 * $\angle ABE = \angle ACF$

Consider the quadrilateral $\Box BFHD$.

We have that $\angle BFH$ and $\angle BDH$ are both right angles.

Thus two opposite angles of $\Box BFHD$ sum to two right angles

So by Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles $\Box BFHD$ is a cyclic quadrilateral.

From Angles in Same Segment of Circle are Equal:
 * $\angle FBH = \angle FDH$.

By similar analysis of quadrilateral $\Box DHEC$, we note that:
 * $\angle HDE = \angle HCE$

But then we have:
 * $\angle FBH = \angle ABE$

and:
 * $\angle HCE = \angle ACF$

Hence it follows that:
 * $\angle FDH = \angle HDE$

demonstrating that $AD$ is the angle bisector of $\angle FDE$.

The same argument applies to $\angle FDE$ and $\angle FED$.

Hence the result.