Just to rephrase in case the question isn't clear: There are n segments on the 2d plane so that every two segments intersects and no two intersection points are alike (there are no intersection points where 3 lines pass). Can another segment intersecting the rest be drawn (and so the new intersections are also distinct)?
Drawing this question on the board, intuitively the answer is "yes", but I can't put the finger on any explanations as to why. I'll note that if the points aren't distinct the answer is no, as you can make an example with a triangle and any line from one of the vertices to a side.
many thanks in advance