2. (mathematics) An arbitrary point $E$ inside the square $A B C D$ with side 1 is connected to its vertices by segments. The points $P, Q, F$ and $T$ are the points of intersection of the medians of triangles $B C E, C D E, D A E$ and $A B E$ respectively. Find the area of quadrilateral PQFT.
Answer: $S=\frac{2}{9}$

Solution: Let's introduce the following notations (see the figure): $K, L$ - midpoints of segments $A E$ and $E C, G$ - midpoint of segment $B E$. $K L$ is parallel to the diagonal $A C$ and is equal in length to half of it. Triangles $B P T$ and $B L K$ are similar with a similarity coefficient $k=\frac{2}{3}$ (property of medians). Therefore, $T P$ is parallel to $K L$ and $A C$. Moreover, $T P=\frac{2}{3} K L=\frac{1}{3} A C$. Similar relationships

![](https://cdn.mathpix.com/cropped/2024_05_06_f912147981c7262840e0g-1.jpg?height=516&width=546&top_left_y=2152&top_left_x=1383)
hold for $P Q, Q F$ and $F T:$

$$
P Q=\frac{1}{3} B D, Q F=\frac{1}{3} A C, F T=\frac{1}{3} B D
$$

Since the diagonals of the square are perpendicular and equal to each other, quadrilateral $P Q F T$ is a square with side length $a=\frac{\sqrt{2}}{3}$ and, consequently, its area $S=\frac{2}{9}$.