MathNet 2014 Question 98

Let $\omega$ be a circle, and let $ABCD$ be a quadrilateral inscribed in $\omega$. Suppose that $BD$ and $AC$ intersect at a point $E$. The tangent to $\omega$ at $B$ meets line $AC$ at a point $F$, so that $C$ lies between $E$ and $F$. Given that $AE = 6$, $EC = 4$, $BE = 2$, and $BF = 12$, find $DA$.

Supplemental exact coordinate figure.