2004 AMC 10B Problems/Problem 16
Contents
Problem
Three circles of radius 
are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?
Solution 1
The situation is shown in the picture below. The radius we seek is 
. Clearly 
. The point 
is the center of the equilateral triangle 
, thus 
is 
of the altitude of this triangle. We get that 
. Therefore the radius we seek is
.
![[asy] unitsize(2cm); pair A=(0,0), B=dir(0)*2, C=dir(60)*2; draw(circle(A,1)); draw(circle(B,1)); draw(circle(C,1)); dot(A); dot(B); dot(C); draw(A--B--C--cycle); pair D=A+dir(210), E=B+dir(-30), F=C+dir(90); draw(circumcircle(D,E,F)); dot(D); dot(E); dot(F); pair S=(A+B+C)/3; dot(S); draw(S--D); draw(S--E); draw(S--F); label("$S$",S,S); label("$A$",A,SE); label("$D$",D,SW); label("$B$",B,NE); label("$C$",C,ENE); [/asy]](http://latex.artofproblemsolving.com/0/5/6/056b92c537632c2d904effcbd77cab0e9d61be72.png)
Solution 2
Using Descartes' Circle Formula, we can assign curvatures to all the circles: , , , and 
(b/c the bigger circle is externally tangent to all the other circles, the radius of the bigger circle is negative). Then, we can solve:
See also
| 2004 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
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