# Proof of a Radon theorem-type claim, related to rays in the plane (Convex geometry)

jjjja
Misplaced Homework Thread moved to the schoolwork forums from a technical forum
I need to show the following thing: Given a collection of 5 rays (half-lines) in the plane, show that it can be partitioned into two disjoint sets such that the intersection of the convex hulls of these two sets is nonempty.

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Is this a homework question? What have you done so far to try to prove it?

jjjja
I need to show the following thing: Given a collection of 5 rays (half-lines) in the plane, show that it can be partitioned into two disjoint sets such that the intersection of the convex hulls of these two sets is nonempty.
Hi, I made this post in a hurry and was unaware of any forum etiquette, so I will explain the problem in more detail now. Sorry for the bad intro into the community.

About the problem, I am solving a homework. For one of the tasks, I am proving something and was given the hint to use this variant of Radon's theorem in one of the steps. I have managed to do the whole proof just assuming that this is true, but for the homework to be complete I need to prove the claim as well. However, I don't even have an idea on how to start it. So let me put it as follows:

Let ##\mathcal{R}=\{r_1,r_2,r_3,r_4,r_5\}## be a collection of rays in the plane. Show that there are disjoint and nonempty sets ##A## and ##B## which partition ##\mathcal{R}## such that ##conv(A)\cap conv(B)## contains a ray.

Thanks for any help!

P.S. The website won't let me preview my LaTeX, so I hope everything formats okay when I post. Sorry if anything messes up.

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