## Task 3 - 130833

In a square $A B C D$ with side length $a$, the side $A B$ is divided into 4 equal segments by the points $A_{1}, A_{2}, A_{3}$, the side $B C$ by the points $B_{1}, B_{2}, B_{3}$, the side $C D$ by $C_{1}, C_{2}, C_{3}$, and $D A$ by the points $D_{1}, D_{2}, D_{3}$. Furthermore, the segments $A_{1} B_{2}, A_{2} B_{3}, B_{1} C_{2}, B_{2} C_{3}, C_{1} D_{2}, C_{2} D_{3}, D_{1} A_{2}$, and $D_{2} A_{3}$ are drawn. Of the intersection points of these segments, the points $P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, P_{6}, P_{7}, P_{8}$ are labeled as shown in the figure.

Calculate the area of the octagon $P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} P_{7} P_{8}$ in terms of $a$!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0663.jpg?height=394&width=419&top_left_y=2093&top_left_x=1361)
One can calculate the area $A_{n}$ of the octagon by subtracting four times the area of the hexagon $A_{2} B B_{2} P_{3} P_{2} P_{1}$ from the area of the square $A B C D$.

The feet of the perpendiculars from $P_{2}$ to $A B$ and $B C$ are $E$ and $F$, respectively. Then $E B F P_{2}$ is a square.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0664.jpg?height=414&width=460&top_left_y=250&top_left_x=250)

Therefore, for the area $A_{8}$ of the octagon, we have

$$
A_{8}=a^{2}-4 \frac{\frac{1}{6} a+\frac{3}{10} a}{2} \cdot\left(\frac{1}{2} a-\frac{3}{10} a\right) \cdot 2+\frac{9}{100} a^{2}=a^{2}-\frac{28}{75} a^{2}-\frac{9}{25} a^{2}=\frac{4}{15} a^{2}
$$

The desired area of the octagon is $\frac{4}{15} a^{2}$.