You are given two points A and B on a plane. How many squares can you draw on the same plane with A and B on the boundary? How many squares can you draw on the plane with A and B as the corners of the square?                                                                                                                                                                                                                                     [Page No. 95]

With A and B only on the boundary: there are infinitely many such squares. We are free to choose the size and the tilt of the square, as long as both points stay on its sides, and there are endlessly many such choices.

With A and B as corners (vertices): the two corners are either adjacent, so AB is a side, or opposite, so AB is a diagonal. If AB is a side, the square can be built on either side of the line AB, giving 2 squares. If AB is a diagonal, this also gives 2 square. So there are 4 squares in all that have A and B as vertices.