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Assume we are given 3 segments in the plane so that there exists a line intersecting them. Regardless of their disjointness, can we always say that there is a line that is tangent to two of them while intersecting the third one?

(Here, a line is "tangent" to a segment if it passes through an endpoint.)

My idea is to take any line that intersects all three and firstly translate then rotate so that it is tangent to two while intersecting the third.

Do you see any flaw in this argument? Is there any counterexample?

Thanks!

Blue
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  • To be clear: Are you defining a line to be "tangent" to a segment if it passes through an endpoint of that segment? – Blue Mar 31 '22 at 12:51
  • Yes, exactly @Blue –  Mar 31 '22 at 13:00
  • The argument could work, but there are some edge cases that you would have to take care of. In particular, what if translating/rotating the line and it keeps ending up tangent to exactly one or exactly 3 of the lines? – Jaap Scherphuis Mar 31 '22 at 13:57
  • Counterexample. $A(0;0)$, $B(1;0)$, $C(2;0)$, $D(0;1)$, $E(1;1)$, $F(2,1)$. Consider segments AF, BE, CD and line $y=0.25$. They satisfy input condition but not claim. – Ivan Kaznacheyeu Mar 31 '22 at 15:32
  • @IvanKaznacheyeu claim says there exists, not any line satisfies the condition. For your example, take the line passing through BF, it is tangent to BE and AF and intersects CD –  Apr 01 '22 at 00:50
  • @MuradAghazada: You're right. I didn't see it. – Ivan Kaznacheyeu Apr 01 '22 at 07:02
  • Ivan's example is exactly one of the trickiest edge cases I was alluding to in my previous comment. If you translate/rotate the line until it passes through at least two vertices, then it either goes through three vertices (ABC or DEF) or coincides with one line segment (CD or AF). To get to the solution you'd have to allow the line pass over a vertex (e.g. A) to get to a solution (the BF solution). This can definitely all be made to work, but I think it is a little tricky to make it watertight and not get bogged down in lots of case work. – Jaap Scherphuis Apr 01 '22 at 15:50

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