Plane Geometry - Bruce E. Shapiro

SECTION 14. BETWEENNESS 63Figure 14.2: Betweenness on a line segment (top), ray (middle), and line(bottom).Definition 14.18 A metric is a function d : P × P ↦→ R that satisfies:1. (∀P, Q ∈ P) d(P, Q) = d(Q, P )2. (∀P, Q ∈ P) d(P, Q) ≥ 03. d(P, Q) = 0 ⇐⇒ P = QThe following theorem is usually also taken as a fourth requirement of ametric in most analysis books.Theorem 14.19 (Triangle Inequality)d(P, Q) ≤ d(P, R) + d(R, Q)We will not be able to prove this theorem until we learn more about planegeometry (see theorem 27.2).Lemma 14.20 Given any two points P, Q ∈ P, there exists a line containingboth P and Q.Proof. Either P = Q or P ≠ Q.If P ≠ Q then there is exactly one line l = ←→ P Q such that P and Q bothline on l (incidence postulate).If P = Q then by the existence postulate there must be a second pointR ≠ P and by the incidence postulate there is a unique line l that containsboth P and R; since P = Q then Q ∈ l.Hence there is a line l that contains both P and Q.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.