AHSME 1991 Problem 19

Triangle $ABC$ has a right angle at $C, AC=3$ and $BC=4$. Triangle $ABD$ has a right angle at $A$ and $AD=12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If $$\frac{DE}{DB}=\frac{m}{n},$$ where $m$ and $n$ are relatively prime positive integers, then $m+n=$ $\text{(A) } 25 \text{(B) } 128 \text{(C) } 153 \text{(D) } 243 \text{(E) } 256$