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I am trying to solve this question regarding the pigeonhole principle:

When 5 points are drawn on the surface of a sphere, 4 of them will lie in the same hemisphere.

After doing some research, all the solutions are similar to the one mentioned in this link:

if there are 5 points on a sphere then 4 of them belong to a half-sphere.

Apparently, I do not understand how dividing the sphere at a great circle formed by any 2 points will cause the rest of the 3 points to be divided among the two hemispheres. E.g. I can draw 5 points on a straight line. If I cut the sphere at the great circle, then all 5 points will be cut into half?

I am certainly misunderstanding something here. Could someone please advise me?

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Donald
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  • I think the claim makes sense if you consider the hemisphere to be closed, i.e. containing the boundary. That way, the circumference of the great circle is contained in both hemispheres and you can say that all the points are in the same hemisphere. – rubik Oct 15 '16 at 08:18
  • Oh. So if the hemisphere is closed, then the points that are cut in half are also included? So does that mean that both hemispheres contain all the points? – Donald Oct 15 '16 at 08:22
  • That's how I interpret it. Otherwise I don't see how it could hold in the extreme case you presented. – rubik Oct 15 '16 at 08:23
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    "the 5 points will be cut into half" : must we take it as humor ? – Jean Marie Oct 15 '16 at 08:31

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