"At the age of 12, I experienced a second wonder of a totally different nature: in a little book dealing with Euclidean plane geometry, which came into my hands at the beginning of a school year. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon me." - Albert Einstein quoted in Pyenson, L. (1985). The Young Einstein: The Advent of Relativity. Boston: Adam Hilger.
The following theorem provides a nice challenge for talented high school learners of geometry especially for those who participate in mathematics competitions. Specifically, it provides practice for proving yet another concurrency (and apparently less known) result.
Theorem
Let D be the first Fermat point1 of △ABC, and E, F, G be the second Fermat points of respectively △ABD, △BCD & △CAD. Then AF, BG and CE are concurrent.
Note: 1 The term 1st 'Fermat point' is used here more generally, not as the unique point in a triangle with no angles greater than 120° that minimizes the sum of the distance to the vertices, but rather as the 1st 'isogonic centre' for any triangle that is obtained by externally constructing equilateral triangles on the sides of a triangle, and is determined by the point of concurrency of the lines connecting the outer vertices of each equilateral triangle with the opposite vertex of the base triangle. Likewise, the 2nd Fermat point here refers to the 2nd isogonic centre obtained by constructing the equilateral triangles to the interior of the base triangle and the corresponding point of concurrency (see Mackay, 1893). At the online Encyclopedia of Triangle Centers these two points are respectively listed as X(13) and X(14).
Another concurrency related to the Fermat point
Explore
1) Drag any of A, B or C to explore.
2) Does your observation hold if △ABC is dragged into the shape of an obtuse triangle or into a degenerate triangle when A, B and C become collinear?
3) Click on the three 'Show △' buttons to show the construction of the equilateral triangles related to the construction of the outer Fermat point E of △ABD.
Challenge
4) Can you explain why (prove that) this theorem is true?
5) Can you explain/prove the theorem in more than one way?
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Free Fermat point Worksheets from Rethinking Proof
In the free PDF download of my Rethinking Proof with Sketchpad book the Fermat-Torricelli point is used for pedagogical purposes to introduce students to the discovery function of proof (De Villiers (1999/2003/2012)- see References below for more specific download details.
Beluhov's Euler Line Concurrency Theorem
From a remarkable theorem proved by Beluhov (2009), the following interesting concurrency also holds in relation to the configuration at the start: the three Euler lines, respectively of △ABD, △BCD & △CAD, are concurrent at the centroid of △ABC.
Submitted Paper
A paper 'Another concurrency related to the Fermat point of a triangle' by myself about the concurrency theorem at the start has been submitted for consideration fot publication to the Mathematics Competitions Journal of the World Federation of National Mathematics Competitions (WFNMC) - all rights reserved.
References
Beluhov, N.I. (2009). Ten Concurrent Euler Lines. Forum Geometricorum, Volume 9, 271–274.
De Villiers, (1999/2003/2012). Guided Worksheets (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.
Note: (Information is provided inside the 'Rethinking Proof' book on where to download the accompanying Sketchpad sketches from, and the Teacher's Notes at the back, provide alternative proofs and generalizations).
Mackay, J.S. (1893). Isogonic Centres of a Triangle. General Report (Association for the Improvement of Geometrical Teaching), Vol. 19 (January), pp. 54-60, The Mathematical Association.
Related Links
Fermat-Torricelli Point Generalizations
Weighted Airport Problem
Bride's Chair Concurrency & Generalization
Napoleon's Theorem: Generalizations & Converses
De Villiers Points of a Triangle
Some Variations of Vecten configurations
A 1999 British Mathematics Olympiad Problem and its dual
Concurrency, collinearity and other properties of a particular hexagon
A side trisection triangle concurrency
Haag Hexagon and its generalization to a Haag Polygon
Haag Hexagon - Extra Properties
Parallel-Hexagon Concurrency Theorem
Toshio Seimiya Theorem: A Hexagon Concurrency result
Some Circle Concurrency Theorems
A variation of Miquel's theorem and its generalization
External Links
Fermat point
The Fermat Point and Generalizations
Fermat Points
X(13) = 1st Isogonic Center (Fermat point, Torricelli point)
X(14) = 2nd Isogonic Center
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Created by Michael de Villiers with WebSketchpad, 10/11 March 2024.