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:
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
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $9$. Conjugate triangles