# Problem 257: Angular Bisectors

![graphic](img257.gif)

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?