# Regular Polygon can be Circumscribed around Circle

## Theorem

Let $P$ be a regular polygon.

Then it is possible to circumscribe $P$ around a circle $C$, whose center is the same as the center of $P$.

## Proof

Let $P$ be a regular polygon.

Aiming for a contradiction, suppose it is not possible to circumscribe $P$ around a circle $C$, whose center is the same as the center of $P$.

Let $AB$, $BC$ and $CD$ be sides of $P$ such that $AB$ and $BC$ are on the tangent to a circle $K$ such that $CD$ is not tangent to that circumference.

There can only be one such circle tangent to both $AB$ and $BC$.

This has to be possible, or all sides of $P$ would be tangent to $K$.

That would make $P$ cyclic which contradicts our supposition about $P$.

Let the center of $K$ be $O$.

As $CD$ is not tangent to $K$, then $OC \ne OD$.

Hence $\angle ABC \ne \angle BCD$ and so $P$ is not equiangular.

Hence a fortiori $P$ is not a regular polygon.

This contradicts our assertion about $P$.

Hence by Proof by Contradiction it follows that $P$ is cyclic.

$\blacksquare$