6. Points $F$ and $G$ are chosen on the sides $AB$ and $BC$ of rectangle $ABCD$, respectively. A perpendicular $FK$ is dropped from point $F$ to side $CD$. A perpendicular $GH$ is dropped from point $G$ to side $AD$. The intersection point of $FK$ and $GH$ is denoted as $E$. Find the area of triangle $DFG$, given that the areas of rectangles $ABCD$ and $HEKD$ are 20 and 8, respectively.
# Solution:

![](https://cdn.mathpix.com/cropped/2024_05_06_821ca558098873554acag-2.jpg?height=402&width=579&top_left_y=1540&top_left_x=1366)

Let $A D=a, D C=b, H D=x$, and $D K=y$.

$$
\begin{aligned}
S_{D F G}=S_{A B C D} & -S_{A F D}-S_{F G B}-S_{D G C}=a b-\frac{1}{2} a y-\frac{1}{2}(b-y)(a-x)-\frac{1}{2} b x \\
& =a b-\frac{1}{2} a y-\frac{1}{2} a y(a b-b x-a y+x y)-\frac{1}{2} b x \\
& =a b-\frac{1}{2} a y-\frac{1}{2} a b+\frac{1}{2} b x+\frac{1}{2} a y-\frac{1}{2} x y-\frac{1}{2} b x=\frac{a b-x y}{2}=\frac{S_{A B C D}-S_{H E K D}}{2}=6
\end{aligned}
$$

Answer: 6