- Is it possible to inscribe a unique ellipse with maximal area into any arbitrary equiangular polygon?

- Is it possible to circumscribe a unique ellipse with minimal area around any arbitrary equiangular polygon?

- Are those inscribed and circumscribed ellipses similar (in the geometric sense)?

E.g. I believe a circle circumscribes, and then maximally inscribes an equiangular triangle;

and, I believe that a unique ellipse inscribes a rectangle, and then a similar ellipse minimally circumscribes the same rectangle.

(I don't know if my two examples are correct so, if they are not, please correct and post a counter examples.)