Characteristic of Interior Point of Circle whose Center is Origin

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

Let $P = \tuple {x, y}$ be a point in the plane of $\CC$.

Then $P$ is in the interior of $\CC$ :
 * $x^2 + y^2 - r^2 < 0$

Proof
Let $d$ be the distance of $P$ from $O$.

Then by definition of interior of $\CC$:


 * $P$ is in the interior of $\CC$ $d^2 < r^2$

and the result follows.