The incenter of a triangle (noch nicht übersetzt)

ABC is an integer sided triangle with incenter I and perimeter p.
The segments IA, IB and IC have integral length as well.

Let L = p + |IA| + |IB| + |IC|.

Let S(P) = ∑ L for all such triangles where p ≤ P. For example, S(10³) = 3619.

Find S(10⁷).