Polar of Point is Perpendicular to Line through Center

Theorem
Let $\CC$ be a circle.

Let $P$ be a point.

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

Then $\LL$ is perpendicular to the straight line through $P$ and the center of $\CC$.

Proof
Let $\CC$ be positioned so as for its center to be at the origin of a Cartesian plane.

Let $P$ be located at $\tuple {x_0, y_0}$.

From Equation of Straight Line in Plane, $P$ can be described as:
 * $y = \dfrac {y_0} {x_0} x$

and so has slope $\dfrac {y_0} {x_0}$.

By definition of polar, $\LL$ has the equation:
 * $x x_0 + y y_0 = r^2$

which has slope $-\dfrac {x_0} {y_0}$.

Hence the result from Condition for Straight Lines in Plane to be Perpendicular.