TITLE: if there are 5 points on a sphere then 4 of them belong to a half-sphere.
QUESTION [37 upvotes]: If there are 5 points on the surface of a sphere, then there is a closed half sphere, containing at least 4 of them.
It's in a pigeonhole list of problems. But, I think I have to use rotations in more than 1 dimension.
Regards

REPLY [71 votes]: Pick two distinct points out of your 5 (if all 5 are identical then they clearly all lie in a single hemisphere).  These two points define at least one great circle (if they're antipodal, they define infinitely many); pick a great circle they define.  This circle then cuts the sphere into two hemispheres.  Now pigeonhole the other three points between these two hemispheres.