On the Fermat point of a triangle

For a given triangle $\triangle ABC$, Pierre de Fermat posed around 1640 the problem of finding a point $P$ minimizing the sum $s_P$ of the Euclidean distances from $P$ to the vertices $A$, $B$, $C$. Based on geometrical arguments this problem was first solved by Torricelli shortly after, by Simpson in 1750, and by several others. Steeped in modern optimization techniques, notably duality, however, we show that the problem \newa{admits a} straightforward solution. Using Simpson's construction we furthermore derive a formula expressing $s_P$ in terms of the given triangle. This formula appears to reveal a simple relationship between the area of $\triangle ABC$ and the areas of the two equilateral triangles that occur in the so-called Napoleon’s Theorem.



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