Problem of the Week

Consider the four points p₁ = (0,0), p₂ = (1,0), p₃ = (0,b), p₄ = (1,b) for b an arbitrary positive real number.  Are there four straight lines, L₁ through p₁, L₂ through p₂, L₃ through p₃, L₄ through p₄, whose points of intesection form the vertices of a square?  Why (not), and what are they (if they exist)?

For the strong of heart:  Consider four arbitrary distinct points in the plane.  Must there always exist four lines, one through each point, whose points of intersection form the vertices of a square?  Why (not), and what are they (if they exist)?

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ã2001 Alberto L. Delgado