Plane Geometry - Bruce E. Shapiro

Section 15The Plane SeparationPostulateIntuitively, we know that a line divides a plane into two halves. These twohalves are called half-planes. We will take this observation as an axiom.Definition 15.1 A set of points S is convex if for every P, Q ∈ S, theentire segment P Q ∈ S.Axiom 15.2 (Plane Separation Postulate) For every line l the pointsthat do not lie on l form two disjoint, convex non-empty sets H 1 and H 2 ,called half-planes, bounded by l such that if P ∈ H 1 and Q ∈ H 2 thenP Q intersects l.More specifically, the plane separation postulate tells us the following (seefigure 15.1).H 1 ∪ H 2 = P − l (15.1)(E, F ∈H 1 ) ⇒H 1 ∩ H 2 = ∅ (15.2)(EF ⊆ H 1 ) ∧ (EF ∩ l = ∅) (15.3)(G, H ∈H 2 ) ⇒(GH ⊆ H 2 ) ∧ (GH ∩ l = ∅) (15.4)P ∈ H 1 ∧ Q ∈ H 2 ⇒ P Q ∩ l ≠ ∅ (15.5)75