## Nested Fermat Quadrilaterals Conjecture

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.

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Nested Fermat Conjecture

This page uses **JavaSketchpad**, a World-Wide-Web component of *The Geometer's Sketchpad.* Copyright © 1990-2008 by KCP Technologies, Inc. Licensed only for non-commercial use.

Michael de Villiers, 30 July 2010.