As I wrote, I have length of the diagonals $10$ and $6\sqrt{2}$ and an angle $45^{\circ}$ between them. What is a radius of this quadrilateral? I have no idea how to approach. I know there is partial answered here:
https://math.stackexchange.com/questions/1989878/radius-of-a-cyclic-quadrilateral-given-diagonals
but I can't see way out from this.
Also it would be interesting to see a construction of such quadrilateral.
Except if the radius of the quadrilateral is something other than the radius of the circumference involved in which case I do not know what it means, I'm afraid the alleged radius is not univocally determined. There are many radius possible (actually an infinity). In the figure below, imperfectly drawn, there are two diagonals D and d, of equal length in both circles and forming an angle of $45^{\circ} $. But the radius are clearly unequal.
