Why is the Fano plane a Projective Plane?

Question

One of the definitions of a projective plane is that there are four points, no three of the points are collinear. But clearly in the Fano Plane, there are 7 points and there exist three points that are collinear.

Fano Plane: https://en.m.wikipedia.org/wiki/Fano_plane

Can someone please explain?

Thanks very much in advance.

Answer 1

In the Fano plane, as you know, there are $7$ points, and $3$ of those $7$ points lie on a line $L$. There are $7-3=4$ points which do not lie on the line $L$, and no three of those four points are collinear. So, yes, there are four points such that no three of them are collinear.