Question 14
Let $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ be a convex hexagon such that $A_{i} A_{i+2} \parallel A_{i+3} A_{i+5}$ for $i=1,2,3$ (we take $A_{i+6}=A_{i}$ for each $i$). Segment $A_{i} A_{i+2}$ intersects segment $A_{i+1} A_{i+3}$ at $B_{i}$, for $1 \leq i \leq 6$, as shown. Furthermore, suppose that $\triangle A_{1} A_{3} A_{5} \cong \triangle A_{4} A_{6} A_{2}$. Given that $[A_{1} B_{5} B_{6}]=1$, $[A_{2} B_{6} B_{1}]=4$, and $[A_{3} B_{1} B_{2}]=9$ (by $[X Y Z]$ we mean the area of $\triangle X Y Z$), determine the area of hexagon $B_{1} B_{2} B_{3} B_{4} B_{5} B_{6}$.
