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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 http://cs.smith.edu/%7Eorourke/MathOverflow/AnnulusCircle.jpg