## Tangent Circles

### Problem 510

Circles A and B are tangent to each other and to line L at three distinct points.

Circle C is inside the space between A, B and L, and tangent to all three.

Let `r`_{A}, `r`_{B} and `r`_{C} be the radii of A, B and C respectively.

Let `S`(`n`) = Σ `r`_{A} + `r`_{B} + `r`_{C}, for 0 < `r`_{A} ≤ `r`_{B} ≤ `n` where `r`_{A}, `r`_{B} and `r`_{C} are integers.
The only solution for 0 < `r`_{A} ≤ `r`_{B} ≤ 5 is `r`_{A} = 4, `r`_{B} = 4 and `r`_{C} = 1, so `S`(5) = 4 + 4 + 1 = 9.
You are also given `S`(100) = 3072.

Find `S`(10^{9}).