Problem of the Week

Consider the four lines, L₁, L₂, L₃, L₄, with respective equations x = 0, x = 1, y = 0, and y = b, for b an arbitrary positive real number.  Are there four points, p₁ on L₁, p₂ on L₂, p₃ on L₃, p₄ on L₄, which form the vertices of a square?  Why (not)?

For the strong of heart:  Consider four arbitrary distinct lines in the plane.  Must there always exist four points, one on each line, which form the vertices of a square?  Why (not)?

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ã2001 Alberto L. Delgado