We know that two distinct points determine a unique line. How many lines are determined by any two of the nine points marked on the given grid?
Solution 1: To count the number of lines, we will divide the lines according to their positions.

- lines parallel to the sides of the squares: three horizontal and three vertical;

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-081.jpg?height=309&width=314&top_left_y=1496&top_left_x=948)

- lines parallel to the diagonals of the squares: $3+3=6$;

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-081.jpg?height=357&width=351&top_left_y=1897&top_left_x=927)

- other lines: we have $4 \times 2=8$ lines, forming a star, as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-081.jpg?height=346&width=348&top_left_y=2351&top_left_x=934)

Thus, in total, we have $3+3+6+8=20$ lines.

Solution 2: Note that the central point is superfluous, as every line passing through it and another point of the grid also passes through a third point of the grid. Therefore, the central point can be eliminated from our considerations. Thus, the problem is reduced to calculating how many lines are determined by any two points of the eight points of the given grid.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-082.jpg?height=240&width=252&top_left_y=671&top_left_x=982)

With these eight points, $8 \times 7$ possible pairs of distinct points can be formed (8 possibilities for the first element of the pair and 7 possibilities for the second). Since the order in which the pair is formed does not affect the line determined by it, this number must be divided by 2, that is,

$$
\frac{8 \times 7}{2}=28
$$

Finally, we notice that some lines are counted three times and, therefore, must be subtracted twice from this number 28. These are the four lines that contain the sides of the square that delimits the grid. Thus, the total number of lines determined by the grid is

$$
\frac{8 \times 7}{2}-2 \times 4=20
$$