Segment arithmetic allows us to complete the chain of logical connections between an abstract geometry satisfying axioms studied in Chapter 2 with the geometries over fields studied in Chapter 3. We will show how to define addition and multiplication of line segments in a Hilbert plane satisfying the parallel axiom (P). In this way, the congruence equivalence classes of line segments become the positive elements of an ordered field F (Section 19). Using this field F we can recover the usual theory of similar triangles (Section 20).
KeywordsLine Segment Positive Element Cartesian Plane Similar Triangle Angle Bisector
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