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CAF123
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Homework Statement
Consider two unshaded circles ##C_r## and ##C_s## with radii ##r>s## that touch at the origin of the complex plane. The shaded circles ##C_1,C_2...C_7## (labeled in counterclockwise direction sequentially) all touch ##C_r## internally and ##C_s## externally. ##C_1## also touches the real axis and ##C_i## and ##C_{ i+1}## touch for ##i=1...6##.
Let ##r_i## denote the radius of ##C_i##. Then show that for ##i=1,2..., ## $$r_i^{-1} + 3r_{i+2}^{-1} = 3r_{i+1}^{-1} + r_{i+3}^{-1}.$$
See picture attached.
Homework Equations
Inversions in the Complex plane
The Attempt at a Solution
I do not really see how to begin this problem and how it may be solved via methods from a Complex Analysis course. I was thinking initially to express the radii in the plane, but then this does not seem to help because it looks difficult to obtain an expression for the circles ##C_4..C_7## given the diagram. We are studying inversion in the complex plane, so that is why I put it in the relevant equations subsection, but I do not see why/if this helps. I then thought about induction since ##i## is an integer, but then to prove the base case would require knowledge of the radii of the circles.
Thanks for any hints.