If a finite set of n>2 points in the plane are not
all on one line, then prove that there exists a line through exactly two of the points.
Wow! This is harder to prove than I would have expected. Or at least, I don't yet have the key insight that proves it for any finite n.
I do notice that this does not appear to be true in spherical geometry. For instance, place 4 points equally spaced around the equator, and one at the North and South Pole. They are not all on a single line (ie, great circle), but there is no line (ie, great circle) that goes through exactly two of them.
Planely true
