1.3. Replace the ellipsis with such a natural number $p$ so that the answer to the following question is unique: how many lines are drawn on the plane, if it is known that they intersect at ... different points?
1.3. Answer. $p=2$.

We will prove that if there are two intersection points, there can only be three lines. Suppose there are only two intersection points $A$ and $B$, and let $l_{1}$ and $l_{2}$ be the lines intersecting at point $A$, and $l_{3}$ be a line not passing through $A$ (such a line exists, otherwise all lines would intersect at $A$). Line $l_{3}$ intersects only one of the lines $l_{1}, l_{2}$, say $l_{1}$, at point $B$, and is parallel to the other line $l_{2}$ (figure a). Verify that no more lines can be drawn without increasing the number of intersection points.

![](https://cdn.mathpix.com/cropped/2024_05_21_20eb9f6ef2c568605a63g-24.jpg?height=322&width=349&top_left_y=481&top_left_x=174)
a)

![](https://cdn.mathpix.com/cropped/2024_05_21_20eb9f6ef2c568605a63g-24.jpg?height=386&width=460&top_left_y=487&top_left_x=558)

Fig. 1.3.

Note that the proof uses Euclid's "fifth postulate" - the axiom stating that through a point, no more than one line can be drawn parallel to a given line.

We will prove that the number of intersection points cannot be different from 2. If there is only one intersection point, the number of lines can be any number starting from 2, provided that all lines pass through this point. If the number of intersection points $n>2$, the number of lines is not uniquely determined. This is illustrated by the example of the arrangement of lines shown in figure b: a line $l_{n+1}$ can be drawn parallel to $l_{n}$, or it can be not drawn, without affecting the number of intersection points.