Comment by tzs
4 hours ago
It can help to think about theorems like this by restating them as a puzzle asking for a counterexample.
Given N points, N > 2, can you arrange them in a Euclidean plane so that (1) they are not all on the same line, and (2) every line that goes through two of the points must also go through at least one more of the points?
The theorem says that you cannot do this.
Well put. A similar way to say it: ask the 9 year old to draw ALL of the possible lines connecting two points in the set — of course this is possible, just might take awhile. The theorem says that at least one of the lines drawn must hit only the two points the 9 year old used, not any others.