This is an interesting, but so far unproved (or refuted) conjecture, I discovered experimentally some 12 years 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 rapidly converge towards a parallelogram, four 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. The real challenge, of course, is to demonstrate the result in a short, elegant way without having to resort to heavy, rather daunting algebra.
Nested Fermat Conjecture
Michael de Villiers, 30 July 2010.