SMO 2009 Senior Round 1 Question 16

In the figure below, $ABC$ is a triangle and $D$ is a point on side $BC$. Point $E$ is on side $AB$ such that $DE$ is the angle bisector of $\angle ADB$, and point $F$ is on side $AC$ such that $DF$ is the angle bisector of $\angle ADC$. Find
$$\frac{AE}{EB}\cdot\frac{BD}{DC}\cdot\frac{CF}{FA}.$$