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.

22 January 2015, Created with GeoGebra

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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