2007 UNCO Math Contest II Problems/Problem 9
Problem
A circle is inscribed in an equilateral triangle whose side
length is 
. Then another circle is inscribed externally
tangent to the first circle but inside the triangle as shown.
And then another, and another. If this process continues
forever what is the total area of all the circles? Express
your answer as an exact multiple of 
(and not as a
decimal approximation).
![[asy] path T=polygon(3); draw(unitcircle,black); draw(scale(2)*T,black); draw(shift(2/sqrt(3),-2/3)*scale(1/3)*unitcircle,black); draw(shift(2/sqrt(3)/3,-2/9)*shift(2/sqrt(3),-2/3)*scale(1/9)*unitcircle,black); [/asy]](http://latex.artofproblemsolving.com/8/a/f/8af6a485cf6e58a89645057e34404328fd5cbaa7.png)
Solution
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See Also
| 2007 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
