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.
The Petr–Douglas–Neumann theorem asserts the following.
In the case of triangles, the value of n is 3 and that of n − 2 is 1. Hence there is only one possible value for k, namely 1. The specialisation of the theorem to triangles asserts that the triangle A1 is a regular 3-gon, that is, an equilateral triangle.
A1 is formed by the apices of the isosceles triangles with apex angle 2π/3 erected over the sides of the triangle A0. The vertices of A1 are the centers of equilateral triangles erected over the sides of triangle A0. Thus the specialisation of the PDN theorem to a triangle can be formulated as follows:
The last statement is the assertion of the Napoleon's theorem.
In the case of quadrilaterals, the value of n is 4 and that of n − 2 is 2. There are two possible values for k, namely 1 and 2, and so two possible apex angles, namely:
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