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**Jacobi's Theorem (1825)**: 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

I'm grateful to Dutch colleague, Floor van Lamoen, for bringing to my attention that the Fermat generalization given above is usually attributed to Karl Friedrich Andreas Jacobi (1795-1855). See p. 26 of his 1825 book Triangulorum Rectilineorum.

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 *Mathematical Gazette*.

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*.

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This page was created by Michael de Villiers in 2009, modified 30 June 2011; 21 January 2019, 14 July 2021; 16 February 2022.