Plane/Lines/Number of intersection points/Example

We consider in the plane ${}E$ a configuration of ${}n$ lines, and we ask ourselves what the maximal number of intersection points of such a configuration might be. It does not make a difference whether we think of the plane as ${}\mathbb {R} ^{2}$ (a Cartesian plane with real coordinates), or simple of a plane in the sense of elementary geometry. The only important thing is that two lines either intersect in exactly one point, or they are parallel. If ${}n$ is small, it is easy to find the answer.

${}n$	${}0$	${}1$	${}2$	${}3$	${}4$	${}5$	${}n$
${}S(n)$	${}0$	${}0$	${}1$	${}3$	${}6$	${}?$	${}?$

But as soon an ${}n$ gets a bit larger (${}n=5,10,\ldots$?), the answer is not so clear anymore, as it gets very difficult to imagine the situation in a precise way. The imagination becomes just a rough idea of many lines with many intersection points, but it is not possible to draw any precise conclusion from this. A useful approach to understand the problem is to understand what may happen when we add a new line to a given line configuration, when instead of ${}n$ lines we consider ${}n+1$ lines. Suppose that, for some reason, we know what the maximal number of intersection points for ${}n$ lines is, maybe we have even a formula for this. If we then can understand how many new intersection points we may get by adding a new line, then we know the maximal number of intersection points for ${}n+1$ lines.

Now, this passage is indeed easy to understand. The new line can at most intersect every old line in exactly one point, therefore at most ${}n$ new intersection points may occur. If we choose the new line in such a way that it is not parallel to any of the given lines (what is possible since there are infinitely many directions) and such that the intersection points of the new line do not coincide with old intersection points (what is possible by possibly taking a line parallel to the direction found), we get exactly ${}n$ new intersection points. Hence, we deduce the (preliminary) formula

${}S(n+1)=1+2+3+\cdots +n-2+n-1+n\,$

or

${}S(n)=1+2+3+\cdots +n-3+n-2+n-1\,,$

so just the sum of the first ${}n-1$ natural numbers.