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Petr–Douglas–Neumann theorem
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.
==Statement of the theorem==
The Petr–Douglas–Neumann theorem asserts the following.〔
:''If isosceles triangles with apex angles 2kπ/n are erected on the sides of an arbitrary n-gon A0, and if this process is repeated with the n-gon 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 An−2 is formed whose centroid coincides with the centroid of A0''.
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