On 4/7/12, rcs@xmission.com <rcs@xmission.com> wrote:
... The final comment about splitting up the sides other than 1 vs N-1 should give a very simple 2-2 formula for quadrilaterals, and could lead to an angle-free formula relating the lengths of the sides and the diagonals. Presumably this would give the 2nd diagonal as a quadratic in the 4 sides and the first diagonal. And there's likely something pretty involving the sides of a pentagon and the "inscribed star".
Rich
The relation between the sides and diagonals of a plane quadrilateral is that the volume of the tetrahedron with those edge-lengths vanishes. The volume is given by the Cayley-Menger determinant for 3-space, a cubic in the squares of the edge-lengths. See eg. http://en.wikipedia.org/wiki/Distance_geometry [I know most of you have heard all this before several times --- Rich obviously wasn't paying attention.] WFL