You are given one point from each side of a square. Reconstruct the square with compass and straightedge.
Connect two opposite points (in this example B and D):
Drop perpendicular from A onto BD:
Mark point E on this perpendicular such that AE = BD. Point E will fall on the side of the square with C (or its extension):
Once you have the orientation of one side of the square (line EC), drop the remaining perpendiculars to find the corners.
Source: problem 20 in https://arxiv.org/pdf/1110.1556v2.pdf