Pigeonhole principle: Five points on an orange

Five points are drawn on the surface of an orange. Prove that it is possible to cut the orange in half in such a way that at least four of the points are on the same hemisphere. (Any points lying along the cut count as being on both hemispheres.)

Solution 1:

Through any two points, there is a great circle that passes through those two points. Such a cut will split the other 3 pigeons — oh, I mean points — among 2 halves.

[You can now handle additional points being on the great circle on your own, I believe.]

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