Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Coxeter, H.S.M., Greitzer, S.L.: Geometry Revisited, vol. 19. Mathematical Association of America, Washington (1996)
Martini, H.: On the theorem of Napoleon and related topics. Math. Semesterber. 43(1), 47–64 (1996)
Grünbaum, B.: Is Napoleon’s theorem really Napoleon’s theorem? Am. Math. Mon. 119(6), 495–501 (2012)
Wetzel, J.E.: Converses of Napoleon’s theorem. Am. Math. Mon. 99(4), 339–351 (1992)
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)
Rigby, J.: Napoleon, Escher, and tessellations. Math. Mag. 64(4), 242–246 (1991)
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)
Gueron, S., Tessler, R.: The Fermat–Steiner problem. Am. Math. Mon. 109(5), 443–451 (2002)
Martini, H., Weissbach, B.: Napoleon’s theorem with weights in n-space. Geom. Dedic. 74(2), 213–223 (1999)
Hajja, M., Martini, H., Spirova, M.: New extensions of Napoleon’s theorem to higher dimensions. Beitr. Algebra Geom 49(1), 253–264 (2008)
Graham, A.: Kronecker Products and Matrix Calculus: With Applications, vol. 108. Horwood, Chichester (1981)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)
Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999)
Boyd, S.P., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
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)
Acknowledgments
We would like to thank Dan P. Guralnik for the numerous discussions and kind feedback. We would also like to express our thanks to Horst Martini for clarifying the use of the term Torricelli configuration and helpful suggestions. This work was funded by the Air Force Office of Science Research under the MURI FA9550-10-1-0567.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Horst Martini.
Rights and permissions
About this article
Cite this article
Arslan, O., Koditschek, D.E. On the Optimality of Napoleon Triangles. J Optim Theory Appl 170, 97–106 (2016). https://doi.org/10.1007/s10957-016-0911-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-016-0911-4