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abelian group AFFINE GEOMETRY affine plane 21 affine restriction affine theorem assume assumptions automorphism axial affinities Axiom A2 axioms A1-A3 axis bijection centre Chapter co-punctal collinear collineation of 21 collineation TT construct Corollary 2.1 defined denote Desargues desarguesian plane dilatation distinct lines distinct points dual elements euclidean geometry exactly one point Exercise exists Figure finite fixed line fixed point group D(O Hence holds true homothety identical implies improper point incident with exactly intersection isomorphism Lemma line incident linear linearly transitive normal subgroup obtain pair Pappus plane parallel pencil perspective collineation plane of order point in common point incident point sets points and lines projective closure projective plane PROOF Let satisfies the axioms Show skew field Statement Suppose Theorem 11 Theorem 9 Theorem of Pappus trace translation plane triangles triplet vector space verify yields