Tangent Circles

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 $S (n) = \sum r_{A} + r_{B} + r_{C}$ , for $0 < r_{A} \leq r_{B} \leq n$ where $r_{A}$ , $r_{B}$ and $r_{C}$ are integers. The only solution for $0 < r_{A} \leq r_{B} \leq 5$ 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 $S (10^{9})$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.