plus Further Generalizations

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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.

1) Drag any of the red vertices to explore. Note that the point of concurrency can lie outside the triangle and the constructed triangles can lie inwardly as well.

Fermat-Torricelli Point Generalization (Jacobi's Theorem) plus other further generalizations

Pierre de Fermat and Evariste Torricelli

**Acknowledgement**

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.

**Some Historical Background & Learning Activities**

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, located at the point in the above sketch where *DC*, *EA* and *FB* meet, *minimizes the sum of the distances from it to each of the vertices*. More-over, for equilateral triangles constructed on the sides of the triangle, the line segments *DC*, *EA* and *FB* are equal - see for example, Mackay (1897).

A useful learning activity to illustrate to students the *discovery function* of proof is to let students start by exploring the concurrency of lines, and equality of the corresponding line segments, when *equilateral triangles* are constructed on the sides of a **right** triangle *ABC* rather the general case. By *explaining why* the result is true, it then becomes obvious from the proof that the triangle need not be a right triangle, and therefore immediately generalizes to any triangle. This learning activity is now available in the free PDF download of my *Rethinking Proof with Sketchpad* book^{1}.

One can also use the Fermat-Torricelli result for determining, for example, the optimal position for building an airport for 3 cities of more or less equal size. A learning activity, with a guided proof for students, is also available in the free PDF download of my *Rethinking Proof with Sketchpad* book^{1}.

**Note**: ^{1}(Information is provided inside the book on where to download the accompanying GSP sketches from).

The Fermat-Torricelli Generalization above is also of potential 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.

**Published Paper**

Download an article of mine from the *Mathematical Gazette* (1995) for a proof of the Jacobi theorem above from A generalization of the Fermat-Torricelli point. An earlier, different proof by Pargeter was also published in 1938 in the *Mathematical Gazette*.

**Some Mathematical Applications**

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

i) Click on the '

ii) Drag

**Further Fermat-Torricelli Generalization**

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

2) Click on the link '

Given a hexagon

3) Click on the link '

4) Explore the figure by dragging

5) Next click on the '

6) Drag

7) However, note that the 'midlines concurrency' result is actually more general, since it is valid for any hexagon with opposite sides parallel. In contrast, the 'hexagon corollary' above does not generalize to any hexagon with opposite sides parallel, and a counter-example is not hard to find using dynamic geometry.

8) For another, different construction of a hexagon with opposite sides parallel and with opposite sides in the same ratio, click Using Dilation to construct a Parallel-hexagon with Opposite Sides in Same Ratio. This construction elegantly explains why a parallel-hexagon is obtained as well as why the concurrency is true.

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(Note that the two images directly above are static. To access the dynamic versions of these sketches use the *WebSketchpad* sketch at the top.)

**Published Paper**

Download an article of mine from the *Mathematical Gazette* (1999) for a proof of the further generalization above from A further generalization of the Fermat-Torricelli point.

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**Some Additional Notes**

As mentioned earlier, 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* from lectures at the Annual Meeting of the Mathematical Association. Below is an example of a physical soap bubble demonstration of a minimal path for a square that was often used by me in lectures at UKZN and at some AMESA conferences.

Lastly, a paper by Park & Flores explores the Fermat-Torricelli point from different perspectives, including an innovative *kinematic method*, which uses a constant relation between two vector functions to find a similar relation between the velocities of the corresponding endpoints of the vectors.

**References**

De Villiers, M. (1995). A generalization of the Fermat-Torricelli point. *Mathematical Gazette*, 79(485), pp. 374-378.

De Villiers, M. (1999). A further generalization of the Fermat-Torricelli point. *Mathematical Gazette*, (March), pp. 14-16.

De Villiers, (1999/2003/2012). Learning Activities (free downloads): The Fermat-Torricelli point (pp. 108-114) and Airport Problem (pp. 115-118) from *Rethinking Proof with Sketchpad* (free download), Key Curriculum Press, Emeryville.

Jacobi, K. F. A. (1825). *Triangulorum Rectilineorum*, p. 26.

Mackay, J.S. (1897). Isogonic Centres of a Triangle. *Proc. Edinburgh Math. Soc.* 15, pp. 100-118.

Park, J. & Flores, A. (2014). Fermat's point from five perspectives. *International Journal of Mathematical Education in Science and Technology*, DOI: 10.1080/0020739X.2014.979894.

**Related Links**

Bride's Chair Concurrency & Generalization

Weighted Airport Problem

De Villiers Points of a Triangle

Van Aubel's Theorem and some Generalizations

Another Construction of a Parallel-hexagon with Opposite Sides in Same Ratio

Napoleon's Theorem: Generalizations & Converses

Easy Hexagon Explorations

Parallel-Hexagon Concurrency Theorem

Toshio Seimiya Theorem: A Hexagon Concurrency result

A side trisection triangle concurrency

A 1999 British Mathematics Olympiad Problem and its dual

Concurrency, collinearity and other properties of a particular hexagon

Haag Hexagon and its generalization to a Haag Polygon

Haag Hexagon - Extra Properties

Conway’s Circle Theorem as special case of Side Divider Theorem

**Some External Links**

The Fermat Point and Generalizations

Fermat point

Jacobi's theorem (geometry)

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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; updated 25/26/30 Nov 2023, 7 Dec 2023.