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 vertices1. 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
Note1An 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 > 120o) is that the three angles surrounding the optimal point are each equal to 120o. 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.