Problem 6. Lёsha's cottage plot has the shape of a nonagon, which has three pairs of equal and parallel sides (see figure). Lёsha knows that the area of the triangle with vertices at the midpoints of the remaining sides of the nonagon is 12 acres. Help him find the area of the entire cottage plot.

![](https://cdn.mathpix.com/cropped/2024_05_06_d5dd184c39392378f787g-2.jpg?height=248&width=436&top_left_y=1589&top_left_x=1501)
Answer: 48 acres.

Solution. Let $D E F G H I K L M$ be the given nonagon, and $A B C$ be the triangle with vertices at the midpoints of the remaining (unmarked) sides. From the condition, it follows that the quadrilateral $D F G E$ is a parallelogram, since its opposite sides $D E$ and $F G$ are equal and parallel. In a parallelogram, the diagonals bisect each other, so points $D, A, G$ lie on the same line. Moreover, $D A=A G$.

Similarly, considering the quadrilaterals $H G I K$ and $K M D L$, we get that point $B$ lies on the line $G K$, point $C$ lies on the line $D K$, and $G B=B K, K C=C D$.

The area of the nonagon $D E F G H I K L M$ is equal to the area of the triangle $D G K$, since triangles $A F G, B I K$ and $C M D$ are equal to triangles $A E D, B H G$ and $C L K$, respectively. Finally, $A, B, C$ are the midpoints of the sides of triangle $D G K$, which means,

![](https://cdn.mathpix.com/cropped/2024_05_06_d5dd184c39392378f787g-2.jpg?height=340&width=577&top_left_y=2126&top_left_x=1302)
its area is four times the area of triangle $A B C$. Thus, the area of the entire plot is

$$
S_{D E F G H I K L M}=4 S_{A B C}=48 \text { acres. }
$$

Answer to variant 2: 32 acres.