Ellipses and Equiangular Polygons

- 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.)

No answer? I really don't know how to go about proving if this is true, (or finding a counter-example.) My experience in college has mainly been with algebra. Is there as way to abstract this problem to one in algebra? I've done some very basic work in algebraic geometry, but I wouldn't know how to express the equiangular shapes as subsets of . I don't think they can be expressed as algebraic sets (like the ellipses can be)...? (And how would I even know if my two algebraic sets represent two similar ellipses?)

By the way, I know this was posted last year, but I still don't have an answer.