On the Fermat point of a triangle

For a given triangle △ABC, Pierre de Fermat posed around 1640 the problem of finding a point P minimizing the sum sP 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 mod- ern optimization techniques, notably duality, however, we show that the problem admits a straightforward solution. Using Simpson’s construction we furthermore derive a formula expressing sP in terms of the given triangle. This formula appearsto reveal a simple relationship between the area of △ABC and the areas of the two equilateral triangles that occur in the so-called Napoleons Theorem.