Problem

2004 AMC 10A Problem 23

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 $B$ ?

$[asy] unitsize(15mm); pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0); draw(Circle(D,2)); draw(Circle(A,1)); draw(Circle(B,8/9)); draw(Circle(C,8/9)); label("$A$", A); label("$B$", B); label("$C$", C); label("$D$", (-1.2,1.8)); [/asy]$

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

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

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Category: AMC 10A
Points: 4
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