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Problem 496

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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 ≠ 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).

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三角形的内心与外心

对于一个三边长都为整数的三角形ABC：
记I是三角形ABC的内心。
记D是直线AI与三角形ABC的外接圆的交点（异于A）。

记 F(L) 为所有满足AC = DI 且 BC ≤ L 的三角形的BC边边长的和。

例如，F(15) = 45，因为只有三边长(BC,AC,AB) = (6,4,5), (12,8,10), (12,9,7), (15,9,16)的三角形ABC才满足条件。

求F(10⁹)。

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