To borrow some formulas you used in previous post, there is:
AC^2 = a^2+b^2-2ab*cosx = c^2+d^2-2cd*cosy
ab cosx - cd cosy = constant u
On the other hand, total area s = ab sinx + cd siny. The goal is to find the condition that leads to max(s).
s^2+u^2= (ab)^2+(cd)^2 + 2abcd sinx siny - 2abcd cosx cosy
After googling certain formulas, the above turns out to be:
(ab)^2+(cd)^2 - 2abcd cos (x+y)
For the value to be maximized, x+y = 180 is the best value. In other words, the quadrilateral fits inside a circle.
