The incenter of a triangle

Published on Sunday, 28th September 2014, 07:00 am; Solved by 163;
Difficulty rating: 85%

Problem 482

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).