Titlebook: Euclidean Geometry and its Subgeometries; Edward John Specht,Harold Trainer Jones,Donald H. Book 2015 Springer International Publishing S

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書(shū)目名稱Euclidean Geometry and its Subgeometries影響因子(影響力)




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Ordering a Line in a Pasch Plane (ORD),pper bound, and lower bound of a subset of an ordered line are developed, as well as the connections between order, segments, and rays. Ordering will assume great importance in later chapters which develop the correspondence between a line and the set of all rational (or real) numbers.

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Neutral Geometry (NEUT),s. Every line is an axis for some reflection. A line of symmetry for a set is a line whose reflection maps that set onto itself. Every angle has a line of symmetry, its angle bisector. Compositions of reflections are isometries, and isometric sets are congruent. These concepts provide access to the

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Free Segments of a Neutral Plane (FSEG),ve the trichotomy property. Addition of free segments is defined, and its elementary properties and interactions with ordering are studied. These developments are sufficient to prove the triangle inequality, and provide a first step toward defining distance on a neutral plane.

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Euclidean Geometry Basics (EUC),xiom to arrive at Euclidean geometry. It explores many well-known elementary results from plane geometry involving parallel lines, perpendicularity, adjacent and complementary angles, parallelograms and rectangles.

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Similarity on a Euclidean Plane (SIM),gs are used to define the similarity of two sets. Similarity is shown to be an equivalence relation, and criteria are developed for similarity of triangles. The chapter concludes with a proof of the Pythagorean Theorem, and a proof that the product of the base and altitude of a triangle is constant.

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Rational Points on a Line (QX), a rational multiple of a point on this line, develops the arithmetical properties of such multiples, and uses these to show the existence of an order-preserving isomorphism between the set of all rational numbers and a subset of the line.