# Problem #2133

 2133 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 $B$? $[asy] size(6cm); defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(Circle(origin, 2)); draw(Circle((-1,0), 1)); draw(Circle((6/9, 8/9), 8/9)); draw(Circle((6/9, -8/9), 8/9)); label("A", (-1.2, -0.2), NE); label("B", (6/9, 7/9), N); label("C", (6/9, -7/9), S); label("D", 2*dir(110), dir(110)); [/asy]$ $\mathrm{(A) \ } \frac{2}{3} \qquad \mathrm{(B) \ } \frac{\sqrt{3}}{2} \qquad \mathrm{(C) \ } \frac{7}{8} \qquad \mathrm{(D) \ } \frac{8}{9} \qquad \mathrm{(E) \ } \frac{1+\sqrt{3}}{3}$ This problem is copyrighted by the American Mathematics Competitions.
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