The [vertices](https://artofproblemsolving.com/wiki/index.php/Vertex) of a $3-4-5$ [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) are the centers of three mutually externally tangent [circles](https://artofproblemsolving.com/wiki/index.php/Circle), as shown. What is the sum of the areas of the three circles?
[Original 2006 12A 13.png](https://artofproblemsolving.com/wiki/index.php/File:Original_2006_12A_13.png)
$\mathrm{(A) \ } 12\pi\qquad \mathrm{(B) \ } \frac{25\pi}{2}\qquad \mathrm{(C) \ } 13\pi\qquad \mathrm{(D) \ } \frac{27\pi}{2}\qquad\mathrm{(E) \ }  14\pi$
Let the radius of the smallest circle be $r_A$, the radius of the second largest circle be $r_B$, and the radius of the largest circle be $r_C$. 
\[r_A + r_B = 3\]
\[r_A + r_C = 4\]
\[r_ B + r_C = 5\]
Adding up all these equations and then dividing both sides by 2, we get,
\[r_A + r_B + r_C = 6\]
Then, we get $r_A = 1$, $r_B = 2$, and $r_C = 3$ Then we get $1^2 \pi + 2^2 \pi + 3^2 \pi = 14 \pi \iff\mathrm{(E)}$