Incenter and Circumcenter of Triangle
Given an integer sided triangle $ABC$:
Let $I$ be the incenter of $ABC$.
Let $D$ be the intersection between the line $AI$ and the circumcircle of $ABC$ ($A \ne D$).
We define $F(L)$ as the sum of $BC$ for the triangles $ABC$ that satisfy $AC = DI$ and $BC \le L$.
For example, $F(15) = 45$ because the triangles $ABC$ with $(BC,AC,AB) = (6,4,5), (12,8,10), (12,9,7), (15,9,16)$ satisfy the conditions.
Find $F(10^9)$.