jmc

By the angle bisector theorem, $AC/CD=AB/BD=12/4=3$. Let $AC=3x$ and $CD=x$.



Then by Pythagoras, $(x+4{)}^2+12^2=(3x{)}^2$. This simplifies to $8x^2-8x-160=0$, which factors as $8(x-5)(x+4)=0$, so $x=5$. Therefore, $AC=3x=\boxed{15}$.