Consider a convex quadrilateral whose diagonals have lengths p,q and whose area is A.
What is the maximum area of a rectangle that circumscribes the given quadrilateral?
Question to Ahmed about the definition of "circumscribe": Are we talking about rectangles that completely contain the quadrilateral? Or is it sufficient that the corners of the quadrilateral are *somewhere* on the 4 lines that define the rectangle? I am not sure the notion of "circumscribe" is so welldefined in the general case...
If this is about the former situation, some cases will have "boundary solutions" for the given maximization problem, therefore I doubt very much there is a nice formula for it.
In the latter situation, the formula must be some function of p, q and the angle of the diagonals, or  thanks to Larry's observation  of p, q and A, as the problem statement suggests.

Posted by JLo
on 20200107 16:16:50 