Product of Distances of Polar and Pole from Center of Circle

Theorem
Let $\CC$ be a circle of radius $r$ whose center is at $O$.

Let $P$ be a point.

Let $\LL_1$ be the polar of $P$ with respect to $\CC$.

Let $\LL_2$ be the line $OP$.

Let $N$ be the point of intersection of $\LL_1$ and $\LL_2$.

Then:
 * $ON \cdot OP = r^2$

Proof
Let $U$ and $V$ be the points where $OP$ intersects $\CC$.


 * Distance-from-center-of-polar.png

From Harmonic Property of Pole and Polar wrt Circle, $\tuple {UV, NP}$ form a harmonic range.

That is: