I'm doing a course in elementary euclidean geometry and I'm studying the properties of inscribed and circumscribed polygons and I've come to the following question:
For which $n$ there exist NON-regular* (1) equilateral circumscribed $n$-gons; (2) equiangular inscribed $n$-gons.
So far, not considering degenerated cases we have $n\ge 3$ and I've managed to excludes triangles and pentagons and find rhombi for (1) and rectangles for (2), however I'm not able to find a general procedure to follow.
I'm allowed to use every elementary geometrical fact (like symmetries, proportions, congruence and so on) and even a little of algebra and Cartesian geometry, however I'd prefer a coordinate-free solution.
EDIT: The definition of a regular polygon is a polygon both equilateral and equiangular, so a non-regular polygon is a polygon which is at least equilateral or equiangular and not both. A circumscribable/inscribable polygon is a polygon which can be circumscribed/inscribed by a circle.