Heh, i take some day to solve this problem. Solution, by Master Herr Nikita Glashenko (Никита Глащенко), because my solution got (i move sqare from 0 to pi/2) TLE (slow code). Do next: take one point and mirror it in front of and sheer from O(0,0). Example: 3 4, -3-4, 4-3,-4 3. And take the equalization of the line catched on  two points(x1,y1) (x2,y2). if size of chord(lying on the line catched on  two points(x1,y1) (x2,y2))=r*sqrt(2)[use approximation 1e-7 for example] in one of the four variants (for four point, what you get), then 'yes' else 'no'.
PS: Division by zero mit uns(with us), you should control it.

Too difficult solution. All you should do for every case - just consider two squares...