3. Given $n$ points on a plane, no three of which lie on the same line. A line is drawn through each pair of points. What is the minimum number of pairwise non-parallel lines among them?
3. The case $n=2$ is trivial, so we will assume that $n \geqslant 3$.

We will specify the arrangement of points for which there will be exactly $n$ pairwise non-parallel lines, which is the set of vertices of a regular $n$-gon. We will prove that the number of non-parallel lines is the same as the number of axes of symmetry.

We will associate with each side and each diagonal its perpendicular bisector (this is an axis of symmetry of the $n$-gon). The perpendicular bisectors of parallel sides and diagonals coincide. The converse is also true: for each axis of symmetry, there will be a side or diagonal perpendicular to it. Considering the cases of even and odd $n$ separately, we can see that a regular $n$-gon has exactly $n$ axes of symmetry.

Now we will prove that for $n \geqslant 3$ points, no three of which lie on the same line, there will always be $n$ distinct pairwise non-parallel lines. It is easy to find $n-1$ lines: take an arbitrary point and consider all lines passing through it. It is more difficult to find one more.

Choose one of the given points as the extreme point. For this, consider their convex hull. In other words, consider the convex polygon containing all the given points, with vertices at some of these points (this is the minimal convex polygon containing all the points).

Let $O$ be a vertex of this convex polygon (Fig. 35), and $A$ and $B$ be the adjacent vertices. Then all rays connecting $O$ with the other points pass inside the angle $A O B$. The line $A B$ intersects all of them and, therefore, can be taken as the $n$-th line.

![](https://cdn.mathpix.com/cropped/2024_05_21_6455f48598646f41a890g-117.jpg?height=272&width=310&top_left_y=760&top_left_x=933)

Fig. 35

Comment. The condition of general position of points cannot be replaced by a weaker condition: "not all points lie on the same line." For example, for the vertices of a regular $2 k$-gon and its center, there will be only $2 k$ pairwise non-parallel lines.