Question 24
Quadrilateral $ABCD$ is inscribed in circle $O$ and has sides $AB=3,$ $BC=2,$ $CD=6,$ and $DA=8.$ Let $X$ and $Y$ be points on $BD$ such that $\dfrac{DX}{BD}=\dfrac{1}{4}$ and $\dfrac{BY}{BD}=\dfrac{11}{36}.$ Let $E$ be the intersection of line $AX$ and the line through $Y$ parallel to $AD.$ Let $F$ be the intersection of line $CX$ and the line through $E$ parallel to $AC.$ Let $G$ be the point on circle $O$ other than $C$ that lies on line $CX.$ What is $XF\cdot XG?$