2004 AMC 12A Problem 19

Circle $A$ has radius $1$ and passes through the center of $D$ while being internally tangent to $D,$ so $D$ has radius $2.$ Place the center of $D$ at the origin, with $A$ centered at $(-1, 0).$ By symmetry, $B$ has center $(x, y)$ and radius $r,$ with $C$ its mirror image across the horizontal axis, so the two congruent circles touch on that axis and $y = r.$ Internal tangency to $D$ gives $x^2 + y^2 = (2 - r)^2,$ and external tangency to $A$ gives $(x + 1)^2 + y^2 = (1 + r)^2.$ Subtracting and using $y = r$ yields $x = \tfrac23$ and $r = \tfrac89.$ The radius of circle $B$ is $\tfrac89.$ Thus, the correct answer is D.