Given four points A, B, C, and D.

Construct a square PQRS such that the points A, B, C, and D lie on lines
PQ, QR, RS, and SP respectively
or
describe the configuration of the four points where the construction is not possible.

Note: Don't confuse lines (of infinite extent) with sides (of finite extent).

(In reply to re: It's all relative (partial spoiler?) by Bractals)

Why yes, I absolutely did not understand the note in the problem.  I guess I answered a different question.  Given that we are talking about lines (not sides), I see that it is possible for all four points to be collinear.  For instance, consider the square with opposite corners at (1,1) and (-1,-1).  The colinear points at (-1,2), (0,1), (1,0) and (2,-1) each lie on one of the 4 lines.

Posted by Steve Herman on 2013-01-08 22:32:50