Consider polygons inscribed in a circle of unit radius.
Call two such polygons "equivalent" if their areas are the same and the sums of squares of the lengths of their edges are the same.
Two angles with vertices on a specified circle have the same measure if they are subtended by the same chord; thus the two quantities – the area and the sum of squares of the side lengths – are determined only by the angles between the diagonals, and there are only as many of those as there are edges.
A vague intuition that came from massaging some trigonometric identities makes me wonder if anything of interest can be said about this equivalence relation or about the angles between diagonals when two polygons are equivalent. Is there anything to that?