The Sylvester–Gallai Theorem

Image: Wikimedia Commons Every finite set of points in the Euclidean plane that is not collinear has a line that passes through exactly two of the points. This proof is by Michigan State University mathematician Leroy Milton Kelly. Consider a set S of points that aren’t all collinear, and define a connecting line to be a line that contains at least two of these points. There must be some point P and connecting line ℓ that are closer together than any other point-line pair in the set. Kelly now proves that ℓ contains only two of the points in S. Assume that this isn’t true; that is, assume that ℓ contains more than two points in S. Then it passes through at least three points in the set. At least two of these must fall on the same side of P′, the perpendicular projection of P on ℓ. Call these two points B and C, with B being closest to P′. If we draw a connecting line 𝓂 that passes through P and C, and draw the perpendicular from B to B′ on 𝓂, then BB′ will be shorter than PP′ (because PP′C and BB′C are similar triangles). This is a contradiction — we’d defined P and ℓ as the point-line pair that are closer together than any other pair in the set. So our assumption that ℓ contains more than two points can’t be true.