Another concurrency related to the Fermat point of a triangle

Another concurrency related to the Fermat point of a triangle plus related results

"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 appears to not be well-known, and provides quite a hard challenge to prove. The other theorems further down related to the same configuration are more suitable challenges for talented high school learners of geometry, especially for those who participate in mathematics competitions.

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 & other related results

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?

Explore More 1
6) Click on the 'Show Ratio Sides' button to display the ratios of the sides and their product.
7) What do you notice?
8) Challenge: Can you explain why (prove) your observation in 6) and 7) above is true?
9) Click on the 'Show another concurrency' button.
10) Challenge: Can you explain why (prove) your observation in 9) above is true?
11) Click on the 'Link to concurrent circles' button to navigate to a new sketch.
12) What do you notice about the three circumcircles of triangles AEB, BFC and CGA, and their corresponding circumcentres X, Y and Z?
13) Make conjectures about your observations in 11) and 12) & explore further by dragging to check if they always hold.
14) Challenge: Can you explain why (prove) your conjectures in 13) above are true?
15) Hint: If stuck, consult De Villiers (2022) below.

Explore More 2
16) Click on the 'Link to angles EFG' button to navigate to a new sketch.
17) What do you notice about the angles of △EFG?
18) Make a conjecture & explore further by dragging to check if it always holds.
19) Challenge: Can you explain why (prove) your conjecture in 18) above is true?
20) Hint: If stuck, consult Egamberganov (2017) below.

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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 and Johan Meyer (University of Free State) about the above results 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, M. (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).
De Villiers, M. (2022). Some Circle Concurrency Theorems. Learning and Teaching Mathematics, No. 33, pp. 34-38.
Egamberganov, K. (2017). A generalization of the Napoleon's Theorem. Mathematical Reflections, no. 3, pp. 1-7.
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.

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; updated 22 July 2024; 19/23 August 2024; 26 Sept 2024; 1 Oct 2024.