Length from an Orthocenter Configuration

In acute triangle $ABC$, points $D$ and $E$ are the feet of the altitudes from points $B$ and $C$ respectively. Lines $BD$ and $CE$ intersect at point $H$. The circle with diameter $DE$ again intersects sides $AB$ and $AC$ at points $F$ and $G$, respectively. Lines $FG$ and $AH$ intersect at point $K$. Suppose that $BC=25$, $BD=20$, and $BE=7$. The length of $AK$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a-b$.