# Are there finitely many points in $\mathbb R^2$ that satisfy this condition?

I was trying to solve a problem and I got curious about this other one because it might give me some intuition:

Are there finitely many points in $$\mathbb R^2$$ such that they do not all lie on a straight line, and such that any straight line passing through two of them also passes through a third?

I tried to construct such points by hand and I couldn't do it. Anybody has an idea?

• My intuition (and just starting from a line and building out) is saying probably not in Euclidean Geometry. If we start with two points we need to add a third to that line, as well as another non-colinear point. That point needs another added point for each of the original points, and it keeps building out. There might be some interesting triangle fractals from just what I've been playing with – wjmccann Jun 12 at 15:43