×

On the optimality of Napoleon triangles. (English) Zbl 1347.51006

Summary: An elementary geometric construction, known as Napoleon’s theorem, produces an equilateral triangle, obtained from equilateral triangles erected on the sides of any initial triangle: The centers of the three equilateral triangles erected on the sides of the arbitrarily given original triangle, all outward or all inward, are the vertices of the new equilateral triangle. In this note, we observe that two Napoleon iterations yield triangles with useful optimality properties. Two inner transformations result in a (degenerate) triangle, whose vertices coincide at the original centroid. Two outer transformations yield an equilateral triangle, whose vertices are closest to the original in the sense of minimizing the sum of the three squared distances.

MSC:

51M15 Geometric constructions in real or complex geometry
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Coxeter, H.S.M., Greitzer, S.L.: Geometry Revisited, vol. 19. Mathematical Association of America, Washington (1996) · Zbl 0166.16402
[2] Martini, H.: On the theorem of Napoleon and related topics. Math. Semesterber. 43(1), 47-64 (1996) · Zbl 0864.51009 · doi:10.1007/s005910050013
[3] Grünbaum, B.: Is Napoleon’s theorem really Napoleon’s theorem? Am. Math. Mon. 119(6), 495-501 (2012) · Zbl 1264.01010 · doi:10.4169/amer.math.monthly.119.06.495
[4] Wetzel, J.E.: Converses of Napoleon’s theorem. Am. Math. Mon. 99(4), 339-351 (1992) · Zbl 0756.51017 · doi:10.2307/2324901
[5] Hajja, M., Martini, H., Spirova, M.: On converses of Napoleon’s theorem and a modified shape function. Beitr. Algebra Geom. 47(2), 363-383 (2006) · Zbl 1116.51013
[6] Rigby, J.: Napoleon, Escher, and tessellations. Math. Mag. 64(4), 242-246 (1991) · Zbl 0758.52014 · doi:10.2307/2690831
[7] Kupitz, Y.S., Martini, H., Spirova, M.: The Fermat-Torricelli problem, part I: a discrete gradient-method approach. J Optim. Theory Appl. 158(2), 305-327 (2013) · Zbl 1292.90282 · doi:10.1007/s10957-013-0266-z
[8] Gueron, S., Tessler, R.: The Fermat-Steiner problem. Am. Math. Mon. 109(5), 443-451 (2002) · Zbl 1026.51009 · doi:10.2307/2695644
[9] Martini, H., Weissbach, B.: Napoleon’s theorem with weights in n-space. Geom. Dedic. 74(2), 213-223 (1999) · Zbl 0920.51008 · doi:10.1023/A:1005052106498
[10] Hajja, M., Martini, H., Spirova, M.: New extensions of Napoleon’s theorem to higher dimensions. Beitr. Algebra Geom 49(1), 253-264 (2008) · Zbl 1142.51015
[11] Graham, A.: Kronecker Products and Matrix Calculus: With Applications, vol. 108. Horwood, Chichester (1981) · Zbl 0497.26005
[12] Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012) · doi:10.1017/CBO9781139020411
[13] Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999) · Zbl 1015.90077
[14] Boyd, S.P., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004) · Zbl 1058.90049 · doi:10.1017/CBO9780511804441
[15] Arslan, O., Guralnik, D., Koditschek, D.E.: Navigation of distinct Euclidean particles via hierarchical clustering. In: Akin, H.L., Amato, N.M., Isler, V., van der Stappen, A.F. (eds.) Algorithmic Foundations of Robotics XI, Springer Tracts in Advanced Robotics, vol. 107, pp. 19-36 (2015) · Zbl 0920.51008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.