It might be useful to keep in mind the proof of Poncelet's porism. You may start with a regular $n$-agon and its circumcircle and incircle (concentric). By applying a projectivity, then an affine transformation, you may send the polygon into a polygon and the circumcircle/incircle couple into a couple of off-centered circles. In this new configuration, every polygonal path with its vertices on the circumcircle and its sides tangent to the incircle closes in $n$ steps. In the $n=4$ case it is pretty simple to construct a bicentric polygon:
Start with a circle $\Gamma_{in}$ and a point $P$ outside of it;
Draw the tangents from $P$ to $\Gamma_{in}$ and let $\theta$ denote the angle between such tangents;
Construct a point $Q$ such that the angle between the tangents to $\Gamma_{in}$ from $Q$ is $\pi-\theta$;
Then the tangents from $P$ and the tangents from $Q$ are the sides of a bicentric quadrilateral inscribed in a circle $\Gamma_{out}$;
$\Gamma_{in},\Gamma_{out}, n=4$ is a Poncelet configuration.
It might be useful to keep in mind the proof of Poncelet's porism. You may start with a regular $n$-agon and its circumcircle and incircle (concentric). By applying a projectivity, then an affine transformation, you may send the polygon into a polygon and the circumcircle/incircle couple into a couple of off-centered circles. In this new configuration, every polygonal path with its vertices on the circumcircle and its sides tangent to the incircle closes in $n$ steps. In the $n=4$ case it is pretty simple to construct a bicentric polygon: