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).
If every three segments from set have common interesecting line, than there exist line passing through all segments from this set
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$\begingroup$
combinatorial-geometry
convex-geometry
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0It would improve your Question if you give the source of (the hard version of) this problem. Perhaps the assumption of finitely many segments can be removed by a compactness argument. – 2017-02-19
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1I found this in several sources (I think it is from some contest), but only one I can find now is this http://www.mathteacherscircle.org/assets/session-materials/TShubin%28Mostly%29Simple%28MostlyArea%29ProblemswithSolutions.pdf Easier version is from article that is not in english. – 2017-02-19
1 Answers
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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.
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0It seems you are right, but this is completely surprising for me, I believed this statement to be true. – 2017-06-10