This is analogous to the maximal area of a quadrilateral with sides a,b,c,d (or a,b,d,c):
sqrt((c + b + a - d) (d - c + b + a) (d + c - b + a) (d + c + b - a)) ---------------------------------------------------------------------, 4
with diagonals
(a d + b c) (b d + a c) (b d + a c) (c d + a b) sqrt(-----------------------) and sqrt(-----------------------). c d + a b a d + b c
Whose formula is this? Brahmagupta's (cyclic case). His formula makes it clear that the area is maximized when the vertices lie on a circle. And what about arbitrary pentagons, etc? Are the maximal areas fixed w.r.t. permuting the sides? If the area is maximized when the vertices lie on a circle, we can freely swap adjacent sides, preserving the chord (and circumscribing) arc, and thus area. --rwg