On the sides $AB$ and $AC$ of a triangle $ABC$ select points $D$ and $E$ respectively, such that $AB = 6$, $AC = 9$, $AD = 4$ and $AE = 6$. It is known that the circumscribed circle of $\vartriangle ADE$ interects the side $BC$ at points $F, G$ , where $BF < BG$. Knowing that the point of intersection of lines $DF$ and $EG$ lies on the circumscribed circle of $\vartriangle ABC$ , find the ratio $BC:FG$.