Geometry
Theorems and Constructions
Published on 20. October 2000
Book
Hardback
256 pages
978-0-13-087121-3 (ISBN)
Description
For undergraduate courses in geometry for Math-ed and Math majors.
College Geometry offers students a deep understanding of the basic results in plane geometry and how they are used. Its unique coverage helps students master Euclidean geometry, in preparation for non-Euclidean geometry.
College Geometry offers students a deep understanding of the basic results in plane geometry and how they are used. Its unique coverage helps students master Euclidean geometry, in preparation for non-Euclidean geometry.
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Content
(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 Poincare Disk. The AAA Theorem in Hyperbolic Geometry. Geometry and the Physical Universe.
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 Poincare Disk. The AAA Theorem in Hyperbolic Geometry. Geometry and the Physical Universe.