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I have the following problem at hand:

We have a collection $\mathcal{C} = \{l_1,\dots, l_n\}$ of $n$ line segments in the plane where each segment is contained in a line through the origin. Assuming that each triple in $\mathcal{C}$ can be intersected by a single line, prove that all segments in $\mathcal{C}$ can be intersected by a single line.

My attempt: First we notice that if two segments are on the same line then all segments are on the same line otherwise we can construct a triple that would contradict the assumption so we WLOG assume that all segments are on different lines.

Now we proceed by induction on $n$. For $n=3$ everything is clear so we continue. Assume that it holds for $3,4,\dots, n-1$. Then for every segment in $\mathcal{C}$ there is a line $p_i$ such that $p_i$ intersects all the segments except $l_i$. Now we should have that $D_0(p_i)$ is an element of intersection of the dual sets of all segments except $l_i$. Or in other words all the dual sets have a nonempty intersection. In the same time I know that dual of $l_i$ has a nonempty intersection with dual of any other two $l_j$ but I am not sure how to finish the proof.

Alternate idea: Let $X_i$ be the set of lines intersecting $l_i$. Then by our assumption each three $X_i$ intersect we can apply Helly theorem.

My problem with both ideas is that they do not use the fact that segments are on lines through the origin and this is definetly not true for line segments in general case.

Notation: $D_0(l)$ is a point $a$ such that $\langle x,a \rangle = 1$ for every $x\in l$.

Klop
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    Have you seen this post ? – YNK Dec 11 '22 at 19:16
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    @YNK I didn't but upon reading I don't understand the correlation? Setups are similar but I think my problem is closer to the Skewer theorem than to the question? – Klop Dec 12 '22 at 21:39
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    I wonder if you can do something similar to the proof of the theorem due to Rey, Pastór, and Santaló. See page 22 in the following slides. The set-up there is very similar to your problem, except that all the line segments are parallel. I believe your intuition is correct that Helly's theorem can be applied. The trick is to associate to each line segment an appropriate convex set, and perhaps the proof of Rey-Pastór-Santaló can be inspiration towards this goal. I currently do not know the solution, but I'd like to hear it if you solve it! – Prism Dec 14 '22 at 09:54
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    @Prism I thik that something like this can be done. First we can notice that all segments need to be inside one half-plane (otherwise it's easy to find a contradiction). Further every line segment l_i will have endpoints $t(x_i,y_i), (t+k)(x_i,y_i)$ for $x_i,y_i$ on the unit circle. Now we can probably characterize the lines intersecting such a segment in terms of $ax+b$ but I am not sure how (I get lost in calculations) and then prove that this set is convex. But maybe there is another easier method, I am not sure... – Klop Dec 14 '22 at 11:54

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