As already said: there is no exact representation of the circle using Bezier curves.

To complete the other answers : for Bezier curve with n segments the optimal distance to the control points, in the sense that the middle of the curve lies on the circle itself, is (4/3)*tan(pi/(2n)).

formula for n segments

So for 4 points it is (4/3)*tan(pi/8) = 4*(sqrt(2)-1)/3 = 0.552284749831.

4 point case