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 $F({10}^9)$.