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?

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter