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**Theorem**: If triangles *DBA*, *ECB* and *FAC* are constructed outwardly (or inwardly) on the sides of any ∆*ABC* so that ∠*DAB* = ∠*FAC*, ∠*DBA* = ∠*EBC* and ∠*ECF* = ∠*FCA*, then *DC*, *EA* and *FB* are concurrent.

Fermat-Torricelli Point Generalization

Pierre de Fermat and Evariste Torricelli

In the special case (originally posed by Fermat in the 1600's and solved a little later 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 my *Rethinking
Proof with Sketchpad* book, which is also available at Amazon, and various other book-sellers.

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.* An earlier, different proof by Pargeter was also published in 1938 in the *MathematicalGazette*.

This theorem easily provides a proof for the result
of the so-called *De
Villiers Points of a Triangle* as illustrated in die diagram below. (Click on the preceding Link to view a dynamic sketch).

**Further Fermat-Torricelli Generalization**: If points *K*, *L*, *M*, *N*, *O* and *P* are constructed on the sides of triangle *ABC* so that * ^{BK}*/

Download an article of mine from the *Mathematical Gazette* (1999) for a proof of the result above from *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*.

*Back to "Dynamic Geometry Sketches"*

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This page was created by Michael de Villiers in 2009, modified 30 June 2011, 21 January 2019.