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 n^{2. }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....