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Fermat-Torricelli Point Generalization

In the special case (originally 1st posed by Fermat & solved by
Torricelli) of the above result, when equilateral triangles are
constructed on the sides of a triangle *ABC* with none of its
angles greater than 120 degrees, then the Fermat-Torricelli point
minimizes the sum of the distances from it to each of the vertices^{1}.
It is therefore a useful theorem for determining the optimal position
for building an airport for 3 cities of more or less equal size. This
is available as a learning activity, with a guided proof that students
find interesting, from *Rethinking
Proof with Sketchpad*

The Fermat-Torricelli Generalization above is also of practical
significance as it can be used to solve a 'weighted' airport problem,
e.g. when the cities have populations of different size. For example,
go to *Weighted
Airport Problem*

Download an article of mine from the *Mathematical
Gazette* (1995) for a proof of the result above from *A
generalization of the Fermat-Torricelli point*

This theorem easily provides a proof for the result
of the *De
Villiers Points of a Triangle*

For a still further generalization (1999) go to *A
Further Generalization of the Fermat-Torricelli Point*

**Note**^{1}An interesting property of the
Fermat-Torricelli point that minimizes the sum of the distances to the
3 vertices of a triangle (with none of its angles > 120^{o})
is that the three angles surrounding the optimal point are each equal
to 120^{o}. A physical model with 3 equal weights hung over
pulleys, connected by thin rope, simulates the minimization of the
distances, and moves to the optimal position when the system
stabilizes. Pictures of such a simulation done with the assistance of Mahomed Bacus on the Edgewood Campus, UKZN are shown below.

Another beautiful way to illustrate the Fermat-Torricelli point is
by using Soap Bubble Geometry. The physical properties of soap film are
such that they would always tend to a shape or position of 'minimal
energy'. It can therefore also be used to model solutions for the
shortest road networks between 3, 4, 5, etc. cities (so-called Steiner
networks). Some soap film solutions are illustrated at *Minimal
Soap Film 'Road' Networks*.

This page was modified by Michael de Villiers, 30 June 2011 with