Different sets of collinear points, each two sets always share a point

Question

This one is actually very simple, but I don't know how to precisely prove it:

If $(A,B,C);(C,D,E);(B,E,F);(A,D,F)$ are $4$ sets of collinear points, is it possible to prove that all these six points collinear?

This is not actually a problem, but it serves as a lemma for one of my other problems.

Answer 1

Here is an approximation of a counterexample:

      a     b           c

              f
                  d
                e

Answer 2

Construct $\triangle ACE$. Let $B$ be a point on $AC$ and $D$ be a point on $CE$. If $F$ is the point of intersection of $AD$ and $BE$. Then the six points satisfy the given conditions but are not collinear.