Intersections of vertical lines with non-vertical lines

Mateusz Kula

Intersections of vertical lines with non-vertical lines

General Mathematics 2022-07-14 v1

Authors: Mateusz Kula

Abstract

We answer negatively a question: if $\mathcal F$ is a family of $n\geqslant 3$ non-vertical, pairwise non-parallel lines on the plane and $\bigcap \mathcal F=\emptyset$ , is there a vertical line $L$ such that $L\cap\bigcup \mathcal F$ has exactly $n-1$ or $n-2$ points?

Cite

@article{arxiv.2207.06230,
  title  = {Intersections of vertical lines with non-vertical lines},
  author = {Mateusz Kula},
  journal= {arXiv preprint arXiv:2207.06230},
  year   = {2022}
}

Comments

I solved the problem while I was participating at the conference in B\k{e}dlewo. The problem is at the elementary level and can be applied to teach duality of points and lines

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