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Problem 257

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Angular Bisectors

Given is an integer sided triangle ABC with sides a ≤ b ≤ c. (AB = c, BC = a and AC = b).
The angular bisectors of the triangle intersect the sides at points E, F and G (see picture below).

The segments EF, EG and FG partition the triangle ABC into four smaller triangles: AEG, BFE, CGF and EFG.
It can be proven that for each of these four triangles the ratio area(ABC)/area(subtriangle) is rational.
However, there exist triangles for which some or all of these ratios are integral.

How many triangles ABC with perimeter≤100,000,000 exist so that the ratio area(ABC)/area(AEG) is integral?

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角平分线

如下是边长均为整数的三角形ABC，其三边长为a ≤ b ≤ c（AB = c，BC = a，AC = b）。
三角形的角平分线和对边分别交于点E、F和G（如下图所示）。

线段EF、EG和FG将三角形ABC分成四个小三角形：AEG、BFE、CGF和EFG。
可以证明，对于这四个小三角形来说，三角形ABC的面积与任一小三角形的面积的比值均为有理数。
甚而还存在一些三角形ABC，使得其中部分或全部的比值均为整数。

在所有边长≤100,000,000的三角形ABC中，有多少个使得三角形ABC的面积与三角形AEG的面积的比值为整数？

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