Geometry

Triangle ABC has a right angle at B.  Point D lies on side \overline{AC} such that CD = 6.  The circle with diameter $\overline{CD}$ intersects $\overline{AB}$ at two distinct points, $E$ and $F,$ with $AE < AF.$  If AE = 6 and DE = 4, then the length $BF$ can be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers.  Find m + n.