Three friends bought a square plot of land and want to divide it into three plots of the same area, as indicated in the figure, because at the corner of the plot marked by $A$ there is a good water source. How far from vertex $C$ of the plot should the boundary points $M$ and $N$ indicated in the figure be?

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-022.jpg?height=291&width=305&top_left_y=1037&top_left_x=1481)
Since the areas of triangles $\triangle A B M$ and $\triangle A D N$ are equal, we have

$$
\frac{1}{2}(B M \times A B)=\frac{1}{2}(N D \times A D)
$$

Since the plot is a square, we have $A B=A D$, so

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-082.jpg?height=343&width=371&top_left_y=882&top_left_x=1411)

that $B M=D N$ and, therefore, the figure $A M C N$ is symmetric with respect to the diagonal $A C$. Thus, the area of triangle $\triangle A C N$ is half the area of triangle $\triangle A D N$. Now, since these triangles have the same height, it follows that $D N=2 N C$ and, by symmetry, $B M=2 M C$. We conclude that the distance from the vertex $C$ of the square to the points $M$ and $N$ must be $1 / 3$ of the length of the side of the square.