Orthocenter of Self-Conjugate Triangle

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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 $\triangle PQR$.

$\blacksquare$


Sources