Problem 257

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 perimeter100,000,000 exist so that the ratio area(ABC)/area(AEG) is integral?

**
These problems are part of
Project Euler
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CC BY-NC-SA 2.0 UK
**