Name: Sheryl Webb
Question: There are hundreds of books and papers that would provide a proof. The result is referred to as Sylvester's Problem. Two proofs are given in Introduction to Geometry by HSM Coxeter, pages 6566 and 181182. Below is the proof that appears on pages 65 and 66 which Coxeter attributes to L. M. Kelley. This proof uses the concept of distance in Euclidean space where the proof on pages 181182 uses only ordered geometry. If n points in the real plane are not on one straight line, then there exists a straight line containing exactly two of the points.The n points P_{1},...,P_{n} are joined by at most 1/2 n(n1) lines P_{1}P_{2}, P_{1}P_{3}, etc. Consider the pairs P_{i}, P_{j}P_{k}, consisting of a point and a joining line which are not incident. Since there are at most 1/2 n(n1)(n2) such pairs, there must be at least one, say P_{1}, P_{2}P_{3}, for which the distance P_{1}Q from the point to the line is the smallest such distance that occurs. Then the line P_{2}P_{3} contains no other point of the set. For if it contained P_{4}, at least two of the points P_{2}, P_{3}, P_{4} would lie on one side of the perpendicular P_{1}Q (or possibly one of the P's would coincide with Q). Let the point be so named that these two points are P_{2}, P_{3}, with P_{2} nearer to Q (or coincident with Q). Then P_{2}, P_{3}P_{1} is another pair having a smaller distance than P_{1}Q, which is absurd.
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