HMMT-Feb 2012 geo, Problem 4

There are circles $\omega_{1}$ and $\omega_{2}$. They intersect in two points, one of which is the point $A$. $B$ lies on $\omega_{1}$ such that $A B$ is tangent to $\omega_{2}$. The tangent to $\omega_{1}$ at $B$ intersects $\omega_{2}$ at $C$ and $D$, where $D$ is the closer to $B$. $A D$ intersects $\omega_{1}$ again at $E$. If $B D=3$ and $C D=13$, find $E B / E D$.

Solution: [diagram] By power of a point, $B A=\sqrt{B D \cdot B C}=4 \sqrt{3}$. Also, $D E B \sim D B A$, so $E B / E D=B A / B D=4 \sqrt{3} / 3$. $\fbox{4 \sqrt{3} / 3}$.

Answer: 4 \sqrt{3} / 3