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In geometry, the Petr–Douglas–Neumann theorem (or the PDN-theorem) is a result concerning arbitrary planar polygons. The theorem asserts that a certain procedure when applied to an arbitrary polygon always yields a regular polygon having the same number of sides as the initial polygon. The theorem was first published by Karel Petr (1868–1950) of Prague in 1908. The theorem was independently rediscovered by Jesse Douglas (1897–1965) in 1940 and also by B H Neumann (1909–2002) in 1941. The naming of the theorem as Petr–Douglas–Neumann theorem, or as the PDN-theorem for short, is due to Stephen B Gray. This theorem has also been called Douglas's theorem, the Douglas–Neumann theorem, the Napoleon–Douglas–Neumann theorem and Petr's theorem. The PDN-theorem is a generalisation of the Napoleon's theorem which is concerned about arbitrary triangles and of the van Aubel's theorem which is related to arbitrary quadrilaterals.
References
- Douglas, Jesse (1946). "On linear polygon transformations". Bulletin of the American Mathematical Society 46 (6): 551–561. doi:10.1090/s0002-9904-1940-07259-3. http://www.ams.org/journals/bull/1940-46-06/S0002-9904-1940-07259-3/S0002-9904-1940-07259-3.pdf. Retrieved 7 May 2012.
- "Petr–Neumann–Douglas Theorem.". From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/Petr-Neumann-DouglasTheorem.html. Retrieved 8 May 2012.
- Stephen B. Gray (2003). "Generalizing the Petr–Douglas–Neumann Theorem on n-gons". American Mathematical Monthly 110 (3): 210–227. doi:10.2307/3647935. http://www.experimentalmath.info/workshop2004/gray-article.pdf. Retrieved 8 May 2012.
- Omar Antolín Camarena. "The Petr–Neumann–Douglas theorem through linear algebra". http://www.matem.unam.mx/omar/notes/petr.html. Retrieved 10 Jan 2018.
