ISBN13: 9780130871213
Edition/Copyright: 01
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Cover: Hardback
Year Published: 2001
Weight: 1.3lbs.
Used Condition: Good/Excellent
Geometry : Theorems and Constructions
by Allan Berele and Jerry Goldman
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College Geometry offers readers a deep understanding of the basic results in plane geometry and how they are used. Its unique coverage helps readers master Euclidean geometry, in preparation for non- Euclidean geometry. Focus on plane Euclidean geometry, reviewing high school level geometry and coverage of more advanced topics equips readers with a thorough understanding of Euclidean geometry, needed in order to understand non-Euclidean geometry. Coverage of Spherical Geometry in preparation for introduction of non-Euclidean geometry. A strong emphasis on proofs is provided, presented in various levels of difficulty and phrased in the manner of present-day mathematicians, helping the reader to focus more on learning to do proofs by keeping the material less abstract. For readers pursuing a career in mathematics.
(NOTE: Each chapter concludes with a Chapter Summary.)
0. Notation and Conventions.
Notation. Constructions.
1. Congruent Triangles.
The Three Theorems. Proofs of the Three Theorems. Applications to Constructions. Applications to Inequalities.
2. Parallel Lines.
Existence and Uniqueness. Applications. Distance between Parallel Lines.
3. Area.
Area of Rectangles and Triangles. The Pythagorean Theorem. Area of Triangles. Cutting and Pasting.
4. Similar Triangles.
The Three Theorems. Applications to Constructions.
5. Circles.
Circles and Tangents. Arcs and Angles. Applications to Constructions. Application to Queen Dido's Problem. More on Arcs and Angles.
6. Regular Polygons.
Constructibility. In the Footsteps of Archimedes.
7. Triangles and Circles.
Circumcircles. A Theorem of Brahmagupta. Inscribed Circles. An Old Chestnut (the Steiner-Lehmus Theorem.) Enscribed Circles. Euler's Theorem.
8. Medians.
Center of Gravity. Length Formulas. Complementary and Anticomplementary Triangles.
9. Altitudes.
The Orthocenter. Fagnano's Problem. The Euler Line. The Nine-Point Circle.
10. Miscellaneous Results about Triangles.
Ceva's Theorem. Applications of Ceva's Theorem. The Fermat Point. Properties of the Fermat Point.
11. Constructions with Indirect Elements.
Constructions with Indirect Elements.
12. Solid Geometry.
Lines and Planes in Space. Dihedral Angles. Projections. Trihedral Angles.
13. Combinatorial Theorems in Geometry.
The Triangulation Lemma. Euler's Theorem. Platonic Solids. Pick's Theorem.
14. Spherical Geometry.
Spheres and Great Circles. Spherical Triangles. Polar Triangles. Congruence Theorems for Triangles. Areas of Spherical Triangles. A Non-Euclidean Model.
15. Models for Hyperbolic Geometry.
Absolute Geometry. The Klein-Beltrami Disk. The Poincaré Disk. The AAA Theorem in Hyperbolic Geometry. Geometry and the Physical Universe.
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