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Incenter and circumcenter of triangle

Published on Sunday, 4th January 2015, 01:00 am; Solved by 235;
Difficulty rating: 50%

Problem 496

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 ≠ D).

We define F(L) as the sum of BC for the triangles ABC that satisfy AC = DI and BC ≤ 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(109).