This is an interesting, but so far unproved (or refuted) conjecture, I discovered experimentally around 1998 or so ago with the aid of Sketchpad. Starting with a (convex) quadrilateral ABCD, construct the four Fermat points of the triangles, ABC, BCD, CDA and DAB, which subdivide the quadrilateral, and then iterate the construction on each new quadrilateral, etc. It then appears that the nested Fermat quadrilaterals gradually seem to converge towards a parallelogram, four iterations of which are shown here. Check visually by dragging any of the vertices A, B, C or D.
This conjecture was also presented in a talk at the Annual Conference of the Mathematical Association from 12-15 April 2003 at the University of East-Anglia, Norwich, UK. It might be interesting to check the conjecture further with Wolfram Mathematica. The real challenge, of course, if true, would be to demonstrate the result in a short, elegant way without having to resort to heavy, rather daunting algebra.
Nested Fermat Conjecture
Related Links
Fermat-Torricelli Point Generalization (aka Jacobi's theorem)
Another concurrency related to the Fermat point of a triangle plus related results
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Created by Michael de Villiers, 30 July 2010 with JavaSketchpad; updated to WebSketchpad, 18 March 2025.