On Circumradius Equations of Cyclic Polygons

In a masterfully written (in german language) thirty pages long paper (and published in 1828 in Crelle’s Journal ) A. F. Möbius studied some properties of the polynomial equations for the circumradius of arbitrary cyclic polygons (convex and nonconvex) and produced a polynomial of degree δn = n2 ( n−1 b(n−1)/2c )− 2n−2 that relates the square of a circumradius (r) of a cyclic polygon to the squared side lengths. He also showed that the squared area rationally depends on r, a1, a2, . . . , an. His approach is based, by a clever use of trigonometry, on the rationalization (in terms of the squared sines ) of the sine of a sum of n angles (peripheral angles of a cyclic polygon ). In this way one obtains a polynomial relating the circumradius to the side lengths squared. These polynomials, known also as generalized Heron r−polynomials, are a kind of generalized