Question 7
Suppose $D$, $E$ are points on the sides $AB$, $AC$, respectively, of a triangle $ABC$, and the following are known: $AB = 6$, $AC = 9$, $AD = 4$, $AE = 6$. Suppose, furthermore, the circumcircle of the triangle $ADE$ intersects the side $BC$ at two points $F$, $G$ and the points $B$, $F$, $G$, $C$ are lined up in this order. If the point of intersection of the lines $DF$ and $EG$ lies on the circumcircle of the triangle $ABC$, find the value of $\frac{FG}{BC}$. Here $XY$ denotes the length of the line segment $XY$ as well.