题目257：角平分线

欧拉计划

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?

题目：

下图给定一个边为 a ≤ b ≤ c 的整边三角形ABC。（AB = c，BC = a，AC = b）。
三角形的角平分线分别交边于点 E，F，G （见下图）。

线段 EF，EG，FG 将三角形 ABC 分割成 4 个小三角形：AEG，BFE，CGF 和 EFG。
可以证明对于这 4 个三角形中的任何一个，ABC 的面积/小三角形的面积的比值为有理数。
然而，存在三角形使得一些或者所有比值为整数。

存在多少周长 ≤ 100,000,000 的三角形ABC，使得比值 ABC 的面积/AEG 的面积为整数？