Intersections of Line joining Conjugate Points with Circle form Harmonic Range

Theorem

Let $\CC$ be a circle.

Let $P$ and $Q$ be conjugate points with respect to $\CC$.

Let $A$ and $B$ be the points of intersection of $PQ$ with $\CC$.

Then $A$ and $B$ are harmonic conjugates with respect to $P$ and $Q$.

Proof

By definition of conjugate points, the polar of $P$ passes through $Q$.

Hence by Harmonic Property of Pole and Polar wrt Circle, $A$ and $B$ are harmonic conjugates with respect to $P$ and $Q$.

$\blacksquare$

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $8$. Reciprocal property of pole and polar