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