If every three segments from set have common interesecting line, than there exist line passing through all segments from this set

Question

There are given lines segments in a plane such that for any three of them there exists a line intersecting them. Prove that there exists a line intersecting all these segments. Perhaps I should use Helly's Theorem, but I have no idea how. In one source I found easier version of this problem (with additional assumptions that set of segments is finite and all segments are parallel), but I can't do that too (I think in this case I should do that by induction and after that apply this result to the infinite case).

Answer 1

I believe this is false.

Consider the four black line segments in this picture:

There is a line intersecting any three of them — the red lines — but I don't believe there is any line intersecting all four segments.