Harmonic Property of Pole and Polar

Theorem

Circle

Let $\CC$ be a circle whose radius is $r$ and whose center is at the origin of a Cartesian plane.

Let $P$ be an arbitrary point in the Cartesian plane.

Let $\LL$ be a straight line through $P$ which intersects $\CC$ at points $U$ and $V$.

Let $Q$ be the point where $\LL$ intersects the polar of $P$.

Then $\tuple {PQ, UV}$ is a harmonic range.

Ellipse

Harmonic Property of Pole and Polar wrt Ellipse