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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
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 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 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/KC = CL/LB = CM/MA = AN/NC = AO/OB = BP/PA, triangles OPD, KLE and MNF are constructed so that ∠DOP = ∠FNM, ∠DPO = ∠EKL, ∠ELK = ∠FMN and triangles LMY, NOZ and PKX are respectively similar to triangles OPD, KLE and MNF, then DY, EZ and FX are concurrent.
View: Click on the link 'Further Fermat-Torricelli Generalization' on the bottom right of the above sketch to navigate to a dynamic, interactive sketch illustrating this interesting, further generalization.
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.
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.
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This page was created by Michael de Villiers in 2009, modified 30 June 2011, 21 January 2019, 14 July 2021.