Melody: "It would have been nice if you had given us the jist of your logic anon :)"
I'll try to explain what I did; I'll use a clock (hour hand).
I set the centers of the circles (A and B) at 9:00 and 3:00, then I added 2 more centers (C and D) at 12:00 and 6:00. The circumferences of these circles have created 4 vertices of the square (W, X ,Y and Z) at 10:30, 1:30, 4:30 and 7:30. I drew the sides and the diagonals of the square, and marked the center of it with an (O).
If you draw an isoceles triangle (BDW), and mark a midpoint of (BD) with an (M), we get a perfect right triangle (BMW).
BW = 10cm, BO = 5cm, BM = sin45o*5 = 3.54cm, BM = MO, MW = 9.35cm, OW = MW-MO = 5.82cm
The diagonal of the square: 2OW = 11.62cm
The side of the square: WX = sin45o*11.62 = 8.22cm
(And finally), the area of the square (WXYZ) is: 67.71cm2.
It would be nice if someone made a graph of all this; if you decided to do it, do not draw the whole circumferences; do them to the point where they cross each other; it's gonna look nicer. Thanks!
I'm exhausted !!!