Plane Geometry - Bruce E. Shapiro

SECTION 13. INCIDENCE GEOMETRY 57Figure 13.4: A Klein Disk. Lines l and n pass through point P , and areboth parallel to line m. The Klein Disk is a model of Hyperbolic Geometry.Axiom 13.7 (Elliptic Parallel Postulate) For every line l and for everypoint P ∉ l there is no line m such that P lies on m and m ‖ l.Three-point geometry and geometry on the sphere satisfy the Elliptic parallelpostulate.Axiom 13.8 (Hyperbolic Parallel Postulate) For every line l and forevery point P ∉ l there are at least two lines m such that P lies on mand m ‖ l.The Klein Disk and Five Point geometry (figure 13.5) satisfy the hyperbolicparallel postulate.Theorem 13.9 If l and m are distinct, nonparallel lines, then there existsa unique point P such that P lines on both l and m.Proof. By hypothesis, l ≠ m and l ̸‖ m. Then by the negation of thedefinition of parallel lines, there is a point P that lines on both l and m.To proove that P is unique, we assume that there is a second point Q ≠ Pthat also lies on both lines as our RAA hypothesis.By incidence axiom 1, there is exactly one line n that contains both P andQ.Since P is on l, then since n is unique, l = n.But since Q is on m, then since n is unique, m = n.Hence l = m. This contradicts the hypothesis that l ≠ m. Hence our RAARevised: 18 Nov 2012 « CC BY-NC-ND 3.0.