There are 6 distinct points, such that there between every four points there are three collinear.
Prove that at least five of these given points belong to one line.

My proof:
If we have points A,B,C,D,E,F then
between ABCD there is one point, lets say point D that doesn't belong to line ABC.
Now between ABDE, AB is not collinear with D so ABE must be collinear.
Between ABDF, ABF is collinear.

So we have ABCEF are collinear.


Is this proof correct?