Napoleon's theorem with weights in \(n\)-space
After a short historical introduction the authors start with Napoleon's theorem (given a triangle \(abc\), erect equilateral triangles externally on its sides, then the circumcenters of these triangles form another equilateral triangle) and Toricelli's solution of Fermat's problem to find a point \(f\) with minimal distance sum \(| f-a|+ | f-b| +f-c|\) from three given points \(a, b, c\). They illustrate these theorems with figures and correct a minor mistake in the book ``What is Mathematics'' (1941; Zbl 0060.12302)] of \textit{R. Courant} and \textit{H. Robbins}. This Fermat problem and the Torricelli configuration were generalized to weighted distances. The authors report on the history of this generalization. Then they generalize it again to \(n\) dimensions and obtain two rather lengthy theorems which are not reproduced here; the reader may look up them in the authors' paper. Even a reader who is not much interested in these generalizations can get very valuable information on the history of the problem field connected with the above-mentioned theorems, by the paper itself and by its references, in particular \textit{H. Martini}'s survey paper in Math. Semesterberichte 43, No. 1, 47-64 (1996; Zbl 0864.51009)].
scientific article; zbMATH DE number 1270434