Six Circles Theorem (Elkies)
What is this about?
A Mathematical Droodle
|What if applet does not run?|
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 rn be the radius of the nth 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 Cn is the incircle of two triangles containing the vertex
The area of Δ A1A2A3 can be given by rs or anhn/2.
In Theorem 2 of , Demir shows that the successive rn'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 = anhn/2, we have
|(1)||(r/rn - 1)(r/rn+3 - 1) = (s - an+1)(s - an+2)/[s(s - an)] = tan2an/2,|
where the last equality is well known to students of elementary trigonometry.
To prove the theorem, replace n by n+3 in (1). Since
Because rn+3 < r, we may cancel the common factor and obtain