Circles $A,B,C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B,C$ are congruent. Circle $A$ has radius $3$ and passes through the center of $D$. What is the radius of circle $B$?

1. Let $O$ be the center of $D$. Since $A$ has radius $3$ and passes through $O$ while being internally tangent to $D$, the radius of $D$ is $6$. Put $O=(0,0)$ and the center of $A$ at $(3,0)$. Let $B,C$ have radius $x$; being congruent and tangent to each other, their centers are $(p,\pm x)$. Tangency to $D$ gives $p^2+x^2=(6-x)^2$, and tangency to $A$ gives $(p-3)^2+x^2=(3+x)^2$. Subtracting, $6p-9=27-18x$, so $p=6-3x$. Then $(6-3x)^2+x^2=(6-x)^2$ gives $9x^2-24x=0$, so $x=\frac83$.
2. The radius of $B$ is $\boxed{\dfrac83}$.