On the three diagonals of a cyclic quadrilateral

If the side lengths of a non-degenerate cyclic quadrilateral are given, but not necessarily in cyclic order, then three diagonal lengths arise in the resulting three cyclic quadrilaterals, just as three possible pairs of supplementary angles arise as opposite vertices, and where the diagonals intersect, in each of the three configurations. We obtain a formula for the sum of the lengths of the three diagonals minus the sum of the four sides which enables us to deduce the geometric inequality that the sum of the side lengths is less than the sum of the lengths of the three diagonals. We obtain another formula when these lengths are replaced by their squares, and this yields a similar inequality. A proof of both formulas is given which uses algebraic geometry, but which proceeds by analysis of degenerate situations. Two alternative proofs of the linear version of the inequality (which implies the quadratic version) are supplied which use trigonometry and Lagrange multipliers respectively. An unusual feature of these results is that they refer not to one configuration, but rather concern three possible configurations.