Fermat Torricelli Similar Polygons Concurrency

fermat-torricelli

The Fermat-Torricelli theorem states that if equilateral triangles DBA, ECB, and FAC are constructed (outwardly or inwardly) on the sides of any triangle ABC, then the lines DC, EA, and FB are concurrent. (Note: A dynamic investigation and guided proof of this theorem is available at: The Fermat-Torricelli Point (Rethinking Proof activity)).

Investigation
1) What about other regular polygons constructed on the sides of a triangle? Such as regular pentagons, regular septagons, etc.? Are the lines from the apex vertices to the opposite vertices of the base triangle still concurrent
The dynamic sketch below shows a triangle with regular pentagons on the sides.
2) What do you notice about the lines from the apex vertices to the opposite vertices of the base triangle? Check your observation by dragging. Is the result still valid if the regular pentagons lie towards the inside? Check by dragging one of the vertices of ABC across an opposite side.
3) Click on the 'Hide apex lines' button, then click on the 'Show centroid lines' button to show the lines from the centroids of the regular pentagons to the opposite vertices of the base triangle. What do you notice? Check by dragging.
4) Click on the 'Link to regular septagons' button. Check to see if the concurrency results above in 2) and 3) also hold in this case.

Fermat Torricelli Similar Polygons Concurrency

5) And what about regular quadrilaterals with an even number of sides such as regular quadrilaterals (squares), regular hexagons, etc. Can we define a corresponding 'apex' point for regular polygons with an even number of sides constructed on the sides of a triangle so that the lines from the 'apex' points to the directly opposite vertices (of the base triangle) are still concurrent?
6) Click on the 'Link to squares' and Click on the 'Link to regular hexagons' buttons to navigate to & explore new sketches. In each case, also click on the 'Hide apex lines' button, then click on the 'Show centroid lines' buttons. What do you notice? Check by dragging.
7) Generalize and formulate your observations in the form of statements.
Note: The point of concurrency of the 'centroid lines' in the case where squares are erected towards the outside on the sides of a triangle is known as the outer Vecten point - if the squares lie towards the inside it is known as the inner Vecten point. The configuration of a triangle with squares constructed on the sides is known as the Vecten configuration or Bride's Chair configuration. It is also a very rich context for many interesting geometric properties as well and has direct connections to several other configurations such as Van Aubel's. Some links are provided below for readers who want to explore these further.

Challenge
8) Can you explain why (prove that) your observations in 7) are true?
Hint: Try using the Similar Isosceles △'s result or Jacobi's theorem in the The Fermat-Torricelli Point activity.

References
De Villiers, M. (1989a). Meetkunde, Meting en Intuisie. Pythagoras, No. 20, pp. 44-45.
De Villiers, M. (1989b). Meetkunde, Verklaring en Insig. Pythagoras, No. 21, pp. 33-38.
De Villiers, M. (1994, 1996, 2009). Some Adventures in Euclidean Geometry (free to download). Lulu Press: Dynamic Mathematics Learning.
De Villiers, M. (1995). A Generalisation of the Fermat-Torricelli Point. The Mathematical Gazette, 9(485) (July), pp. 374-378.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.

External Links
Fermat point (Wikipedia)
Fermat Points (Wolfram MathWorld)
The Fermat Point and Generalizations (Cut The Knot)
Vecten points (Wikipedia)
Bride's Chair (Cut The Knot)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Michael de Villiers, created with WebSketchpad, 13-14 Feb 2026.