Square from Four Points, One on Each Side
Solution 1

Here's a problem from an old Russian problem collection by D. O. Shklyarsky, N. N. Chentsov, Y. M. Yaglom Selected Problems and Theorems from Elementary Mathematics, Part 2 (Planimetry) (1952, #70):

There are four points in the plane. Construct a square such that each side of the square (or its extension) would pass through exactly one of the given points.

The applet below illustrates one of the constructions.

Join, say A and C (this is assuming that A and C lie on the opposite sides of the future square). Drop a perpendicular from B to AC and find E such that BE = AC. D and E lie on the same side of the square. The rest of the construction consists in dropping perpendiculars to form the other three sides.

If E happens to coincide with D, the number of solutions is infinite (the problem is not well defined). Otherwise, E could be located on both sides from B, and any of points B, C, D could be thought to lie on the side opposite the one that contains A. Therefore, in general, there are six solutions.

The problem is Part (a) of a tripartite sequence.

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