Among all convex quadrilaterals with given sides(and their order), which one does have maximal sum of diagonals? Does it have any other interesting properties?
I thought about something like this: let $A = (0; 0), C = (x; 0)$ (i. e. fix the length of one diagonal), then we can find points $B$ and $D$, so the desired sum is a function of $x$ with a lot of roots, and we can just find its maximum. But:
- Derivative is very cumbersome.
- We need to ensure not only that $ x \le \min(a + b, c + d)$ but also that resulting quadrilateral is convex, I'm not sure what limits it puts on x.
Any thoughts? Maybe it is just cyclic quadrilateral, but it doesn't look so.