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 A₁, A₂, ..., 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 c = λ + iμ, then to construct C, draw a line through (1 - λ)A + λB perpendicular to AB and mark a point at the distance μ|B - A| from AB.

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 { α₀ α₁ ... α_n-1}. Let K denoted the simplest circulant {0 1 0 ... 0}. Then Kⁿ = I (a unit matrix) and

M = α₀I + α₁K + α₂K² + ... + α_n-1K^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 Kⁿ - 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 ωⁿ = 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 (1 - c_d) + tc_d = (t - ω_d)/(1 - ω_d).

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 - c₁) + c₁t)...((1 - c_n-1) + c_n-1t) = 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 A₁...A_n into the centroid of points A₁, ..., A_n.

Omit now any of the constituent LPTs. Let it be (1) with c_d for some d. (n - 2) LPTs remain. Since LPTs commute, the polygon obtained after (n - 2) steps has the property that a single application of (1) with c_d shrinks it to a point. Which exactly means that the polygon is either regular or star-shaped.

For n = 3, we get the inner or outer Napoleon triangles depending on whether c₁ or c₂ is omitted.

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