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Axioms and coordinate systems for the affine
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0IJ coordinates 1-1 correspondence ABC coordinates affine plane algebraic allowable coordinate system anti-parallel assigns Axiom Chapter circle collinear complex number congruent contains convert convex sets corollary to Theorem cosine definition denote determined diagram distinct points dyadic angles dyadic rationals equation equivalent expression ax fact Figure formulas geometry given half-planes inside point intersection isometry law of cosines least upper bound Lemma length line reflections linear expression mathematics measure nates non-collinear points null angle one-dimensional coordinate system ordered pair parallel projection perpendicular projection point reflection polygon positive number PQ coordinates principal p-measure Proof Prove quadrant angle radian rectangle rectangular coordinate system right angle right triangle scale factor segment sensed angle set of points shows similarity similitude sine standard position straight angle terminal side Theorem 11 tion triangle ABC two-dimensional coordinate system upper sum vertex vertices whence xu yi zero