Plane Geometry - Bruce E. Shapiro

SECTION 13. INCIDENCE GEOMETRY 59R ∉ m, else P, R, S would be collinear, and we have assumed otherwise.Hence l and m are distinct lines that contain P .Theorem 13.12 Let l be a line. Then there exist two distinct lines m andn that intersect l.Proof. Let l be a line.By incidence axiom 2 there are two points P, Q thatlie on l.By theorem 13.10 there exists a third point R that does lie on l.By incidence axiom 1, there exist lines m = ←→ P R and n = ←→ QR.Since P ∈ l and P ∈ m, l intersects m (definition of intersection).Since Q ∈ l and Q ∈ n, l intersects n (definition of intersection).Since R ∉ l then any line that contains R is different from l. The two linesm and n contain R, hence m ≠ l and n ≠ l.Suppose m = n. By definition of m, P ∈ m. By defintion of n, Q ∈ n.Hence if m = n, Q ∈ m, i.e., both P and Q line on m. Hence m = l bythe uniqueness part of incidence axiom 1. This contradicts the previousparagraph, m ≠ l. Hence the assumption m = n must be wrong.Hence m ≠ n, which means there are two distinc lines that intersect l.Theorem 13.13 Let P be a point. Then there exists at least one line thatdoes not contain P .Theorem 13.14 There exist three distinct lines such that no point lies onall three of them.Theorem 13.15 Let P be a point. Then there exist points Q and R suchthat P, Q, R are non-collinear.Proof. Let P and Q be points such that Q ≠ P . By incidence axiom 1,there is a unique line l that contains P and Q.By theorem 13.10 there exists at least one point R that does lie on l. Thepoints P, Q, R are non-collinear.Theorem 13.16 Let P ≠ Q be points. Then there exists a point Q suchthat P, Q, R are non-collinear.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.