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Parallels to an Equilateral Triangle

Be forewarned...this may not seem to be an easy problem...but it is do-able with some nice discoveries along the wayside...


Prove or Disprove: Given any set of three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.


 


Hint: Use GSP to draw some sets of three paralell lines and try to draw equilateral triangles as requested. Any patterns emerge?

For example, focus on the side with vertices anchored on the outermost parallel lines...does it stay a constant length?

But, even if the claim appears true, the task remains for proving the general case...which is possible!

 


Solution Commentary: First, I suggest you browse the student solutions found at nRich website. The one I like the best is the geometrical argument provided by Hyeyoun Chung...age 14!

A related dymanic exploration and solution is offered by Alexander Bogomolnv on his great Cut-the-Knot websit, who also mentions that the problem's original source is I. M. Yaglom's Geometric Transformations I (MAA, 1962, Chapter 2, Problem 18)... a neat book to explore if you have not seen it.

Or, M.J. (Bellingham) notes: "Isn't the answer obvious if you consider rolling the triangle between the two outer parallel lines and follow the 3rd vertex in between? The middle vertex must cover all inner lines." Aha!