Rich wrote: << I think one way of expressing the hard problem is "given a set of edge-lengths {Ei}, determine the radius of the circumscribing circle".
Now it makes sense. Lessee -- a chord of length E in a circle of radius R subtends a central angle ang(E) = 2*arcsin(E / 2R). Since the total central angle subtended by all edges of a cyclic quadrilateral is 2 pi [see below *], it follows that when R is the circumradius, we have (1) pi = f(R) = arcsin(E_1 / 2R) + ... + arcsin(E_n / 2R) given the edge lengths as E_1,..., E_n. Now f(R) is clearly monotonic, so there is a unique value of R making (*) true. (1) is by no means a closed formula for R, but at least it determines R uniquely. It appears that if the formulae for the sine of the sum of n angles is sufficiently cooperative, this could lead to a closed-form expression for R -- which in turn would lead to a closed-form expression for the area of a cyclic n-gon with edges {E_i}. _____________________ * Appropriate adjustments need to be made when the center of the circumcircle does not lie in the polygon's interior. --Dan