Given 5 points on a sphere, show that 4 of them are in the same
hemishpere.
Assume that the hemisphere includes the great circle that forms its boundary.

Pick any two of the points and draw the great circle that passes through them.
If there's another point on the great circle, then we can add that point and
either of the two remaining points.
Otherwise, there's three points not on the great circle.
One of the two regions defined by the great circle must contain two points, and we
can take those two plus the two on the great circle.