The Incenter of a Triangle

$A B C$ is an integer sided triangle with incenter $I$ and perimeter $p$ .
The segments $I A$ , $I B$ and $I C$ have integral length as well.

Let $L = p + | I A | + | I B | + | I C |$ .

Let $S (P) = \sum L$ for all such triangles where $p \leq P$ . For example, $S (10^{3}) = 3619$ .

Find $S (10^{7})$ .

Write-up coming later

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