Plane Geometry - Bruce E. Shapiro

78 SECTION 15. THE PLANE SEPARATION POSTULATETheorem 15.12 (Pasch’s Theorem) Let △ABC be a triangle and supposethat l is a line that does not include A, B, or C. Then if l intersectsAB then it also intersects either BC or CA.Proof. Suppose l intersects AB and does not include any of the verticesA,B, or C.Figure 15.3: Pasch’s Theorem. Any line that intersects AB must alsointersect either AC or BC.BACLet H 1 and H 2 be the two half planes determined by l. Then the points Aand B are in opposite half-planes by the plane separation postulate and thehypothesis. Suppose A ∈ H 1 and B ∈ H 2 (this is just notation; we couldhave made the alternative assignment without any loss of generality).Then either C ∈ H 1 or C ∈ H 2 .If C ∈ H 1 , then B and C are in opposite half planes so BC intersects l, bythe plane-separation postulate.Alternatively, if C ∈ H 2 , then A and C are in opposite half-planes, so ACintersects l, also by the plane separation postulate.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012