Sylvester-Gallai Theoremcite: For any $n$ points in $\RR^{2}$, not all collinear, there exists a line passing through exactly two of them.

Proof: Pick any $2$ points and draw a line $\ell$ through them: suppose a $3$rd point lies on the line (else we are done) and pick the closest point $p\notin\ell$ to the line, at a distance $\delta$
say.

Of our $3$ points in $\ell$, a pair lies on one side of $p$: draw a line $\ell’$ through $p$ and the furthest of the pair from $p$. The distance $\delta’$ between $\ell’$ and the second point is less than $\delta$.

R7arosh

cite. https://math.stackexchange.com/questions/699002 ↩