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Given two concentric circles $[C_1,C_2 ]$ with radii $(R_1 < R_2) $ creating an annulus; where should a third circle ($C_3$, radius $R_3$) be located such that the area of intersection between the annulus and $C_3$ is maximum?

When I was trying to solve the problem I assumed WLOG, due to symmetry, that the center of $C_3$ is placed at some point along the x-axis $(d,0)$ and found $d$ to satisfy

$2(R_3^2 + d^2) = R_1^2 + R_2^2$

(If $d$ is not real, then let $d=0$ and all three circles are concentric)

  1. Is that correct?
  2. If so, is there a geometrical reason for that criteria?
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closed as too localized by Igor Rivin, Ian Agol, Chris Godsil, Gerry Myerson, Andy Putman Apr 11 '12 at 2:39

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

It looks plausible. [It works for a few extremal cases at least.] How did you arrive at that? – Pat Devlin Apr 10 '12 at 0:36
This site is for questions of research interest. I'm not convinced there's any research interest here, in which case may be a better place to post. And I don't see where this is algebraic geometry. – Gerry Myerson Apr 10 '12 at 11:07

Your condition on $d$ ensures that the intersections $a$ and $b$ between circles $C_3$ and $C_1$ and $C_2$ respectively, are at the same $y$-coordinate. Then the different slopes of $C_1$ and $C_2$ at these points ensure that the area is a local maximum.
    Annulus & Circle

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There is a formula for the area of the intersection of two circles of given radii in terms of the distance between the centers. The formula can be found here:

If the radius of $C_3$ is smaller than $R_2-R_1$ or greater than $R_2$ then the answer is obvious. In the other cases you can also calculate the explicit area formula of the intersection. If we denote $\mathcal{A}(C_i,C_j)$ the area of the intersection of the circles $C_i,C_j$. Then the area of the intersection of $C_3$ with the annulus is $\mathcal{A}(C_2,C_3)-\mathcal{A}(C_1,C_3)$, and using the formulas presented in the link you can write the exact formula in terms of $d$, and then optimize with respect to $d$ the formula you get.

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