Much to my surprise this construction must be possible - never mind how I placed the points A, P, Q.

The point B must be on a semi-circle over AP, the point C must be on a semi-circle over PQ and the point D must be on a semi-circle over AQ.

But unfortunately I haven't found yet a second condition to determine

- the length of the square's side OR

- the midpoint of the square OR

- the angle (or the angle OR

- ???

Sorry.