TOPICS
Search

Petr-Neumann-Douglas Theorem

If isosceles triangles with apex angles 2kpi/n are erected on the sides of an arbitrary n-gon A_0, and if this process is repeated with the n-gon A_1 formed by the free apices of the triangles, but with a different value of k, and so on until all values 1<=k<=n-2 have been used in arbitrary order, then a regular n-gon A_(n-2) is formed whose centroid coincides with the centroid of A_0.

Napoleon's theorem and van Aubel's theorem are special cases of the Petr-Neumann-Douglas theorem.

See also

Douglas-Neumann Theorem, Isosceles Triangle, Napoleon's Theorem, van Aubel's Theorem

Portions of this entry contributed by Floor van Lamoen

Explore with Wolfram|Alpha

References

Baker, H. F. "A Remark on Polygons." J. London Math. Soc. 17, 162-164, 1942.Chang, G. "A Proof of Douglas and Neumann by Circulant Matrices." Houston J. Math. 8, 15-18, 1982.Chang G. and Davis, P. "A Circulant Formulation of the Napoleon-Douglas-Neumann Theorem." Linear Alg. Appl. 54, 87-95, 1983.Douglas, J. "Geometry of Polygons in the Complex Plane." J. Math. Phys. Mass. Inst. Tech. 19, 93-130, 1940.Gray, S. B. "Generalizing the Petr-Douglas-Neumann Theorem on n-gons." Amer. Math. Monthly 110, 210-227, 2003.Neumann, B. H. "Some Remarks on Polygons." J. London Math. Soc. 16, 551-560, 1941.Neumann, B. H. "A Remark on Polygons." J. London Math. Soc. 17, 165-166, 1942.Pech, P. "The Harmonic Analysis of Polygons and Napoleon's Theorem." J. Geometry Graphics 5, 13-22, 2001.Petr, K. "Ein Satz über Vielecke." Arch. Math. Physik 13, 29-31, 1908.Wong, Y. C. "Some Extensions of the Douglas-Neumann Theorem for Concentric Polygons." Amer. Math. Monthly 75, 470-482, 1968.

Referenced on Wolfram|Alpha

Petr-Neumann-Douglas Theorem

Cite this as:

van Lamoen, Floor and Weisstein, Eric W. "Petr-Neumann-Douglas Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Petr-Neumann-DouglasTheorem.html

Subject classifications