The Torricelli Point

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In 1650, Fermat proposed the problem of finding a point such that the sum of its distances from three given points is minimal. Torricelli suggested a solution, asserting that the three circles circumscribing equilateral triangles constructed on the sides of and outside the triangle formed by the given points intersect in the minimizing point. This point is called the Torricelli point. Cavalieri showed in 1647 that the line segments from the three given points to the Torricelli point meet at 120° to each other. Simpson asserted and proved in 1750 that the lines joining the outside vertices of the exterior equilateral triangles defined above to the opposite vertices of the given triangle intersect in the Torricelli point. These three lines are called the Simpson lines. Heinen in 1834 proved that the lengths of the three Simpson lines are the same and are equal to the sum of distances from the Torricelli point to the three given points.

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When one of the angles in the given triangle is at least 120°, the Torricelli point lies outside of the given triangle and is no longer the minimizing point. The minimizing point is in this case the vertex of the obtuse angle. Courant and Robbins, in their famous book What Is Mathematics?, referred to this problem as the Steiner problem. The popularity of their book has been the main reason that the misnomer "The Steiner problem" has stuck, but it's also the main reason that interest in this problem has spread.

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


Snapshots


Details

F. K. Hwang, D. S. Richards, and P. Winter, "The Steiner Tree Problem," Annals of Discrete Mathematics, 53, 1992.



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