Maximum area of intersection between annulus and circle? 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)


*

*Is that correct? 

*If so, is there a geometrical reason for that criteria?

 A: 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.
   

A: 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: http://mathworld.wolfram.com/Circle-CircleIntersection.html
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.
