--description--

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 $\frac{\text{area}(ABC)}{\text{area}(\text{subtriangle})}$ 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 $\frac{\text{area}(ABC)}{\text{area}(AEG)}$ is integral?

--hints--

angularBisectors() should return 139012411.

assert.strictEqual(angularBisectors(), 139012411);

--seed--

--seed-contents--

function angularBisectors() {

  return true;
}

angularBisectors();

--solutions--

// solution required