What is AD?

From Ceva’s Theorem, we have

$\dfrac{AF}{FB}\cdot \dfrac{BD}{DC}\cdot \dfrac{CE}{EA}=1$

$\dfrac{4}{2}\cdot \dfrac{BD}{DC}\cdot \dfrac{1}{3}=1$

$\dfrac{BD}{DC}=\dfrac{6}{4}$

that is,

$\dfrac{BD}{DC}=\dfrac{AB}{AC}$ .

It now follows from the converse to the Angle Bisector Theorem that $AD$ is the angle bisector.