Steiner Networks for Four Points

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The Fermat (or Torricelli) point of a triangle minimizes the sum of the distances from to each of the vertices.

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Steiner extended this problem by considering more than three points and asking for the shortest route connecting all of them. The solution for four suitably located points can be found by constructing an equilateral triangle on opposite sides of a quadrilateral. For this network to be the shortest, it is necessary that the two green vertices lie within the convex hull of the four original red vertices.

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Contributed by: Jaime Rangel-Mondragon (March 2011)
Open content licensed under CC BY-NC-SA


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Details

D. Wells, The Penguin Dictionary of Curious and Interesting Geometry, London: Penguin, 1992.



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