let's have a polygon with the vertexes over a circle of radius R.
prove that the sum of the squares of the lenghts of the sides and diagonals is lower or equal than the square of the radius times n2.
and when the equality holds?
some exploration shows that the equality holds when the polygon is regular, but I have no general proof.
I would say that some trigonometry could help, and also something about quadratic average, but.....no idea how to proceed....