Orthocenter of Self-Conjugate Triangle

Theorem
Let $\CC$ be a circle.

Let $\triangle PQR$ be a self-conjugate triangle with respect to $\CC$.

Then the orthocenter of $\triangle PQR$ is the center of $\CC$.

Proof
By definition of self-conjugate triangle:
 * $PR$ is the polar of $Q$
 * $QR$ is the polar of $P$

and from Self-Conjugate Triangle needs Two Sides to be Specified:
 * $PQ$ is the polar of $R$

all with respect to $\CC$.

Let $O$ be the center of $\CC$.

Then from Polar of Point is Perpendicular to Line through Center:
 * $PQ \perp OR$
 * $QR \perp OP$
 * $PR \perp OQ$

That is: $OP$, $OQ$ and $OR$ are lines perpendicular to the sides of $\triangle PQR$ which all pass through $O$.

Hence, by definition, $O$ is the orthocenter of $\triangle PQR$.