Douglas' Theorem
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I shall follow the articles by B.H.Neumann (1942) and J.Douglas (1940). (K. Petr published a proof of the result discussed below in 1905.) The theorem's most recent designation is PDN, Petr-Douglas-Newmann, giving credit to all three authors.)
Let the vertices of an n-gon be ordered A_{1}, A_{2}, ..., A_{n} and considered as complex numbers. We shall apply a certain operation to every pair of consecutive vertices.
For any two complex numbers A and B, the all-important operation is
(1) | C = (1 - c)A + cB |
If c is real, points C lie on a straight line through A and B. For complex c, A,B, and C form a triangle similar to the triangle formed by points 0, 1, and c. The orientation is preserved. If
When applied to all pairs of successive vertices of an n-gon P, operation (1) yields another polygon, P_{c}. In geometric terms, P_{c} consists of the apexes of similar triangles erected on the sides of the polygon P. Vertices of P_{c} are generated successively from those of P. Jesse Douglas calls (1) a linear polygonal transformation - LPT. LPT has several important properties. First, P and P_{c} are concentric, i.e. share the centroid.
(2) | ∑P_{i} = ∑(P_{c})_{i} |
To obtain other properties, arrange vertices of P and P_{c} as n-dimensional vectors. (1) gets then represented as a linear transformation with an n×n circulant matrix L_{c} with the first row given by {(1 - c) c 0 ...0}. A general circulant matrix M can also be defined by its first row {
α_{0}
α_{1} ...
α_{n-1}}. Let K denoted the simplest circulant
M = α_{0}I + α_{1}K + α_{2}K^{2} + ... + α_{n-1}K^{n-1}
or M = p_{M}(K). p_{M}(t) is known as the auxiliary polynomial of M. When two circulants are multiplied, their auxiliary polynomials are multiplied modulo K^{n} - I. This shows that any two circulant matrices commute, and so are LPTs. From here we also obtain a fact that is not quite obvious from geometric considerations, viz., the result of a sequence of LPTs does not depend on the order of individual transformations.
The auxiliary polynomial of a product of several LPTs is simply a product of the terms ((1 - c) + ct). For the generalization of Napoleon's theorem we are going to select LPTs in a particular manner.
Let ω_{d} denote (n - 1) distinct roots of unity ω^{n} = 1, excluding 1. In other words, ω_{d}'s are all roots of the equation
(3) | r(t) = t^{n-1} + t^{n-2} + ... + t + 1 = 0 |
For d = 1, 2, ..., n-1, define
(4) | c_{d} = 1/(1 - ω_{d}) |
Verify that c_{d} = (1 + i·cot(πd/n))/2 and also
With thus selected c_{d}'s, (1) is equivalent to a construction of isosceles triangles with simple apex angles. (Note that for n = 3, the angles are ±120°). From (3) and (4) we have
((1 - c_{1}) + c_{1}t)...((1 - c_{n-1}) + c_{n-1}t) = r(t)/r(1) = r(t)/n
which corresponds to the circulant matrix {1/n 1/n 1/n... 1/n}. This matrix transforms any vector-polygon A_{1}...A_{n} into the centroid of points A_{1}, ..., A_{n}.
Omit now any of the constituent LPTs. Let it be (1) with c_{d} for some d.
For n = 3, we get the inner or outer Napoleon triangles depending on whether c_{1} or c_{2} is omitted.
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