Pedal Circle of Incenter is Incircle

Theorem

Let $\triangle ABC$ be a triangle whose incenter is $H$.

Then the pedal circle of $H$ is the incircle of $\triangle ABC$.

Proof

Let $\CC$ denote the incircle of $\triangle ABC$.

Aiming for a contradiction, suppose $\CC$ is not the pedal circle of $H$.

By definition, the sides of $\triangle ABC$ are tangent to $\CC$.

Hence from Line at Right Angles to Diameter of Circle: Porism, the line through $H$ to those point of tangency to $\CC$ are perpendicular to the sides.

Hence $\CC$ is the pedal circle of $H$ by definition.

Hence by Proof by Contradiction, the pedal circle of $H$ is the incircle.

$\blacksquare$

Also see

• Pedal Circle of Excenter is Excircle