Square from Four Points, One on Each Side
Solution 2

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.

18 January 2013, Created with GeoGebra

Choose two pairs of points, say, A, B and C, D, that lie on adjacent sides of the square to be constructed. Draw two circles CAB and CCD, on AB and CD as diameters. Two vertices of the square lie on those circles - actually on certain pair of semicircles, and those semicircle ought to have the "same orientation". Thus we have two choices for a pair of semicircles (the applet picks up just one such pair). The line through the two vertices passes through the midpoints (G and H) of the remaining semicircles. Finding the intersection of that line with the selected semicircles gives two vertices; the other two are found next.

If G and H happen to coincide there are infinitely many solutions; otherwise, there are 6: 3 choices for an adjacent pair of vertices times two choices of similarly oriented semicircles.

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

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