Circles $A, B$ and $C$ are externally tangent to each other, and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$. What is the radius of circle $B$? $\mathrm{(A) \ } \frac23 \qquad$ $ \mathrm{(B) \ } \frac {\sqrt3}{2} \qquad$ $ \mathrm{(C) \ } \frac78 \qquad$ $ \mathrm{(D) \ } \frac89 \qquad$ $ \mathrm{(E) \ } \frac {1 + \sqrt3}{3}$
Solution
Circle $D$ has radius 2, assume $B$ has radius $x$.
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 $B$?
$\mathrm{(A) \ } \frac23 \qquad$ $ \mathrm{(B) \ } \frac {\sqrt3}{2} \qquad$ $ \mathrm{(C) \ } \frac78 \qquad$ $ \mathrm{(D) \ } \frac89 \qquad$ $ \mathrm{(E) \ } \frac {1 + \sqrt3}{3}$ 
