Bisectors

Here is a solution that doesn't require any complex geometry. We see that we know all 3 side lengths of triangle BID. We know a formula that can tell us the area directly knowing 3 sides of the triangle, Heron's formula.

First we calculte the semi-perimeter $\frac{3+4+6}{2}=\frac{13}{2}$. 

Then we apply Heron's formula. $Area = \sqrt{\frac{13}{2}(\frac{13}{2}-3)(\frac{13}{2}-4)(\frac{13}{2}-6)}$. Simplifying, we get $Area = \sqrt{\frac{455}{16}}=\frac{\sqrt{455}}{4}$ 

So, the Area is sqrt(455)/4 which is around 5.3327.