Reciprocal Property of Pole and Polar

Theorem
Let $\CC$ be a circle.

Let $P$ and $Q$ be points in the plane of $\CC$.

Let $\PP$ and $\QQ$ be the polars of $P$ and $Q$ with respect to $\CC$ respectively.

Let $Q$ lie on $\PP$ with respect to $\CC$.

Then $P$ lies on $\QQ$.

That is:
 * if $P$ lies on the polar of $Q$, then $Q$ lies on the polar of $P$
 * if the pole of $\PP$ lies on $\QQ$, then the pole of $\QQ$ lies on $\PP$

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

Let $P = \tuple {x_0, y_0}$.

Let $Q = \tuple {x_1, y_1}$.

The polar of $P$ is given by:
 * $x x_0 + y y_0 = r^2$

The polar of $Q$ is given by:
 * $x x_1 + y y_1 = r^2$

Let $Q$ lie on the polar of $P$.

Then $Q$ satisfies the equation:
 * $x_0 x_1 + y_0 y_1 = r^2$

which is exactly the same as the condition for $P$ to lie on $Q$.

Also see

 * Definition:Conjugate Points (Geometry)
 * Definition:Conjugate Lines