Geometry Theorems and Constructions

by Berele, Allan; Goldman, Jerry

• ISBN13: 9780130871213
• ISBN10: 0130871214
• Edition: 1st
• Format: Paperback
• Copyright: 2000-10-06
• Publisher: Pearson

Summary

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.

Table of Contents

Preface	xi
Notation and Conventions	1	(3)
Notation	1	(1)
Constructions	2	(2)
Congruent Triangles	4	(17)
The Three Theorems	4	(2)
Proofs of the Three Theorems	6	(4)
Applications to Constructions	10	(4)
Application to Inequalities	14	(7)
Chapter Summary	18	(3)
Parallel Lines	21	(9)
Existence and Uniqueness	21	(2)
Applications	23	(3)
Distance between Parallel Lines	26	(4)
Chapter Summary	28	(2)
Area	30	(19)
Area of Rectangles and Triangles	30	(4)
The Pythagorean Theorem	34	(3)
Area of Triangles	37	(3)
Cutting and Pasting	40	(9)
Chapter Summary	47	(2)
Similar Triangles	49	(10)
The Three Theorems	49	(3)
Applications to Constructions	52	(7)
Chapter Summary	56	(3)
Circles	59	(23)
Circles and Tangents	59	(3)
Arcs and Angles	62	(4)
Applications to Constructions	66	(2)
Application to Queen Dido's Problem	68	(3)
More on Arcs and Angles	71	(11)
Chapter Summary	77	(5)
Regular Polygons	82	(10)
Constructibility	82	(3)
In the Footsteps of Archimedes	85	(7)
Chapter Summary	90	(2)
Triangles and Circles	92	(17)
Circumcircles	92	(2)
A Theorem of Brahmagupta	94	(1)
Inscribed Circles	95	(3)
An Old Chestnut (the Steiner-Lehmus Theorem)	98	(1)
Escribed Circles	99	(3)
Euler's Theorem	102	(7)
Chapter Summary	106	(3)
Medians	109	(9)
Center of Gravity	109	(2)
Length Formulas	111	(1)
Complementary and Anticomplementary Triangles	112	(6)
Chapter Summary	116	(2)
Altitudes	118	(12)
The Orthocenter	118	(2)
Fagnano's Problem	120	(2)
The Euler Line	122	(2)
The Nine-Point Circle	124	(6)
Chapter Summary	127	(3)
Miscellaneous Results about Triangles	130	(14)
Ceva's Theorem	130	(3)
Applications of Ceva's Theorem	133	(2)
The Fermat Point	135	(3)
Properties of the Fermat Point	138	(6)
Chapter Summary	142	(2)
Constructions with Indirect Elements	144	(4)
Constructions with Indirect Elements	144	(4)
Solid Geometry	148	(31)
Lines and Planes in Space	148	(7)
Dihedral Angles	155	(4)
Projections	159	(4)
Trihedral Angles	163	(16)
Chapter Summary	171	(8)
Combinatorial Theorems in Geometry	179	(12)
The Triangulation Lemma	179	(1)
Euler's Theorem	180	(3)
Platonic Solids	183	(2)
Pick's Theorem	185	(6)
Chapter Summary	189	(2)
Spherical Geometry	191	(20)
Spheres and Great Circles	191	(1)
Spherical Triangles	192	(3)
Polar Triangles	195	(4)
Congruence Theorems for Triangles	199	(3)
Areas of Spherical Triangles	202	(2)
A Non-Euclidean Model	204	(7)
Chapter Summary	206	(5)
Models for Hyperbolic Geometry	211	(14)
Absolute Geometry	211	(4)
The Klein-Beltrami Disk	215	(2)
The Poincare Disk	217	(2)
The AAA Theorem in Hyperbolic Geometry	219	(1)
Geometry and the Physical Universe	220	(5)
Chapter Summary	222	(3)
Hints to Starred Exercises	225	(4)
Index	229