[math-fun] polygons

Carrying Dan Asimov's analysis a litle further, the edges Ek fit a circumcircle of radius R (diameter D=2R) when the product Prod ( sqrt(D^2-Ek^2) + i Ek ) has 0 imaginary part, i.e. is real. This gives a messy algebraic expression for D, and hints that there might not be a closed form when the number of edges is increased, because we run afoul of unsolvable quintics etc. The full permutability of the edges is another hint in that direction. Of course there's nothing that says finding D is equivalent to finding the area of the polygon. Knowing D, we can indeed get the area as the sum of triangles of sides R,R,Ek; but perhaps the reverse direction is more difficult. Rich rcs@cs.arizona.edu