G4.4 In figure $1, A B C D$ is a rectangle, and $E$ and $F$ are points on $A D$ and $D C$, respectively. Also, $G$ is the intersection of $A F$ and $B E, H$ is the intersection of $A F$ and $C E$, and $I$ is the intersection of $B F$ and $C E$. If the areas of $A G E, D E H F$ and CIF are 2, 3 and 1 , respectively, find the area of the grey region $B G H I$. (Reference: 2014 FI1.1)
Let the area of $E G H=x$, area of $B C I=z$, area of $B G H I=w$
Area of $B C E=\frac{1}{2}$ (area of $\left.A B C D\right)=$ area $A D F+$ area $B C F$
$$
\begin{array}{l}
x+w+z=3+x+2+1+z \\
\Rightarrow w=6
\end{array}
$$
$\therefore$ Area of the grey region $B G H I=6$
Remark: there is a spelling mistake in the English version. old version: ... gray region ...