Sylvester–Gallai theorem

Remarks

A line that contains exactly two points of $S$ is called an ordinary line. Kelly's proof considers all pairs of a point $p \in S$ and a line $\ell$ through two points of $S$ with $p \notin \ell$. It takes a pair with the least distance from $p$ to $\ell$. Suppose $\ell$ contained three points of $S$. Then two of them would lie on the same side of the foot of the perpendicular from $p$, or one of them at the foot. This gives a pair with a smaller positive distance, a contradiction; so $\ell$ is ordinary. The hypothesis that $S$ is finite is needed: the integer lattice $\Z^2$ has no ordinary line. The theorem also fails over the complex numbers. The nine inflection points of a smooth complex cubic curve form a configuration in which every line through two of them contains a third. For example, the four vertices of a square have ordinary lines: each side contains exactly two of the vertices. Green and Tao proved that for all sufficiently large $n$, a set of $n$ points not all on a line has at least $n/2$ ordinary lines.

Going deeper

For a student

The Sylvester–Gallai theorem is a starting point of combinatorial incidence geometry, like the Szemerédi–Trotter theorem. A corollary is that $n$ points not all on a line determine at least $n$ distinct lines. Its failure over $\C$ and over finite fields shows that it depends on the order structure of the real numbers.