From the Fermat points to the De Villiers3 points of a triangle

The article starts with a problem of finding a point that minimizes the sum of the distances to the vertices of an acute-angled triangle, a problem originally posed by Fermat in the 1600’s, and apparently first solved by the Italian mathematician and scientist Evangelista Torricelli. This point of optimization is therefore usually called the inner Fermat or Fermat-Torricelli point of a triangle. The transformation proof presented in the article was more recently invented in 1929 by the German mathematician J. Hoffman. After reviewing the centroid and medians of a triangle, these are generalized to Ceva’s theorem, which is then used to prove the following generalization of the Fermat-Torricelli point from [3]: “If triangles DBA, ECB and FAC are constructed outwardly (or inwardly) on the sides of any ∆ABC so that ∠DAB =∠CAF , ∠ DBA = ∠ CBE and ∠ ECB = ∠ ACF then DC, EA and FB are concurrent.”However, this generalization is not new, and the earliest proof the author could trace is from 1936 by W. Hoffer in [1], though the presented proof is distinctly different. Of practical relevance is the fact that this Fermat-Torricelli generalization can be used to solve a “weighted” airport problem, for example, when the populations in the three cities are of different size. The author was also contacted via e-mail in July 2008 by Stephen Doro from the College of Physicians and Surgeons, Columbia University, USA, who was considering its possible application in the branching of larger arteries and veins in the human body into smaller and smaller ones. On the basis of an often-observed (but not generally true) duality between circumcentres and in centres, it was conjectured in 1996 [see 4] that the following might be true from a similar result for circumcentres (Kosnita’s theorem), namely: The lines joining the vertices A, B, and C of a given triangle ABC with the incentres of the triangles BCO, CAO, and ABO (O is the incentre of ∆ABC), respectively, are concurrent (in what is now called the inner De Villiers point). Investigation on the dynamic geometry program Sketchpad quickly confi rmed that the conjecture was indeed true. (For an interactive sketch online, see [7]). Using the aforementioned generalization of the Fermat-Torricelli point, it was now also very easy to prove this result. The outer De Villiers point is similarly obtained when the excircles are constructed for a given triangle ABC, in which case the lines joining the vertices A, B, and C of a given triangle ABC with the incentres of the triangles BCI1, CAI2, and ABI3 (Ii are the excentres of ∆ABC), are concurrent. The proof follows similarly from the Fermat-Torricelli generalization.