2004 AMC 12A Problems/Problem 19

Problem 19

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 $B$ ?

$\text {(A)} \frac23 \qquad \text {(B)} \frac {\sqrt3}{2} \qquad \text {(C)}\frac78 \qquad \text {(D)}\frac89 \qquad \text {(E)}\frac {1 + \sqrt3}{3}$

Solution

$[asy] unitsize(20mm); pair A=(0,1),B=(-8/9,-2/3),C=(8/9,-2/3),D=(0,0), E=(0,-2/3); draw(Circle(D,2)); draw(Circle(A,1)); draw(Circle(B,8/9)); draw(Circle(C,8/9)); draw(A--B--C--A); draw(B--D--C); draw(A--E); dot(A);dot(B);dot(C);dot(D);dot(E); label("$D$", D,N); label("$A$", A,N); label("$B$", B,W); label("$C$", C,E); label("$E$", E,S); label("$1$",(-.4,.7)); label("$1$",(0,.5),E); label("$r$", (-.8,-.1)); label("$r$", (-4/9,-2/3),S); label("$h$", (0,-1/3),E); [/asy]$