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.
$\blacksquare$