Six Circles Theorem (Elkies)
What is this about?
A Mathematical Droodle

Explanation

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The applet is supposed to illustrate the following problem:

The problem has been posed by Noam D. Elkies in the American Mathematical Monthly (1987, 877). A solution by Jiro Fukuta has been published in 1990 (v. 97, issue 6, pp. 530-531)

Let r be the radius of the incircle of Δ A₁A₂A₃, s be its semiperimeter, and r_n be the radius of the n^th circle. We may chose indices so that C₁, C₂ are companion incircles generated by a line through A₃, with C₁ in the triangle containing A₃A₁ and C₂ in the triangle containing A₃A₂. Then, successively, the circle C_n is the incircle of two triangles containing the vertex A_{n mod 3}. Thus we define A_n by A_n = A_m , whenever n = m (mod 3). Let a_n be the angle at A_n, let a_n be the length of the side opposite A_n, and let h_n be the altitude from A_n. We have

The area of Δ A₁A₂A₃ can be given by rs or a_nh_n/2.

In Theorem 2 of [1], Demir shows that the successive r_n's satisfy

Multiplying by r and subtracting it from 1, we obtain

Multiplying the corresponding equations for n and n+2 and dividing by the equation for n+1 yields

Since rs = a_nh_n/2, we have 1 - 2r/h_n = (s - a_n)/s, so this equation becomes

where the last equality is well known to students of elementary trigonometry.

To prove the theorem, replace n by n+3 in (1). Since a_n+3 = a_n, we have

Because r_n+3 < r, we may cancel the common factor and obtain

which suffices.

References

1. H. Demir, Incircles within, Math. Magazine 59 (1986) 77-83

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