# 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 $\CC$ if and only if:

$x^2 + y^2 - r^2 < 0$

## Proof

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

 $\ds d$ $=$ $\ds \sqrt {\paren {x - 0}^2 + \paren {y - 0}^2}$ Distance Formula $\ds \leadsto \ \$ $\ds d^2$ $=$ $\ds x^2 + y^2$

Then by definition of interior of $\CC$:

$P$ is in the interior of $\CC$ if and only if $d^2 < r^2$

and the result follows.

$\blacksquare$