Squares_through

Squares through

Given four points A,B,C,D, construct a square, whose sides pass through these points. The construction is based on the following remark: From B draw the orthogonal line to AC and take B* such that BB* is equal to AC. The point B* is on the side of a square that solves the problem. Once two points (B* and D) are known the whole square is determined. B** which is symmetric to B* with respect to B, gives another solution.
Doing the permutations (A)(B)(C,D) and (A,C)(B)(D) and repeating the constructions, we get the six solutions dislpayed in Squares_through_4_points.html

It is much easier to do the complete construction, using the scheme-sockets tool. For this copy-paste the above scheme into the document. Catch the copy and move it appart to distinguish from the original. Then do 2 times a plug of the copy into the original by corresponding the master-points A, B, C, D according to the permutations indicated in the text-box on the top:
The first plug associates the masters of the original with those of the copy in the following way: A- >A, B- >B, C- >D, D- >C. The second time the correspondence is A- >C, C- >A, B- >B, D- >D. After the second plug, and hidding the many auxiliary lines involved we get the scheme of Squares_through_4_points.html .

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