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:

$ON \cdot OP = r^2$

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

That is:

$\ds \dfrac {VN} {NU}$

$=$

$\ds -\dfrac {VP} {PN}$

$\ds \leadsto \ \ $

$\ds \dfrac {ON} {OV}$

$=$

$\ds \dfrac {OV} {OP}$

as $O$ is the midpoint of $UV$

$\ds \leadsto \ \ $

$\ds ON \cdot OP$

$=$

$\ds OV^2$

$\ds $

$=$

$\ds r^2$

$UV$ is the diameter of $\CC$

$\blacksquare$