Complex Numbers and Geometry ( Spectrum Series) » (1st Edition)

Liang-shin Hahn was born in Tainan, Taiwan. He received his B.S. from the National Taiwan University and his Ph.D. from Stanford University. After a brief period of teaching at the Johns Hopkins University, he moved to the University of New Mexico where he has been ever since. He has held visiting positions at the University of Washington (Seattle), the National Taiwan University (Taipei), the University of Tokyo, the International Christian University (Tokyo) and Sophia University (Tokyo), giving his the distinction of teaching mathematics in three countries, and in three languages.
 
As an unabashed admirer of the late George Polya, the author is found of telling students: "The trick in teaching mathematics is that I do the easy part and you do the hard part," because the author s motto in heuristic teaching is: "Don t try to teach everything. Teach the basic ideas, then use questions to guide students to explore and discover for themselves."
 
The author has posed many interesting problems in The American Mathematical Monthly, and his conjecture on Egyptian fractions is widely cited. He has been solely responsible for composing the New Mexico Context problems since 1990. He is also the co-author, with Bernard Epstein, of Classical Complex Analysis.
 
He enjoys playing ping-pong, cultivating roses, listening to classical music and solving as well as creating mathematical puzzles.

This book demonstrates how complex numbers and geometry can be blended together to give easy proofs of many theorems in plane geometry.

Mathematical Reviews

Provides a self-contained introduction to complex numbers and college geometry written in an informal style with an emphasis on the motivation behind the ideas.... The author engages the reader with a casual style, motivational questions, interesting problems and historical notes.

Preface
1 Complex Numbers
1.1 Introduction to Imaginary Numbers
1.2 Definition of Complex Numbers
1.3 Quadratic Equations
1.4 Significance of the Complex Numbers
1.5 Order Relation in the Complex Field
1.6 The Triangle Inequality
1.7 The Complex Plane
1.8 Polar Representation of Complex Numbers
1.9 the nth Roots of 1
1.10 The Exponential Function
Exercises

2 Applications to Geometry
2.1 Triangles
2.2 The Ptolemy-Euler Theorem
2.3 The Clifford Theorems
2.4 The Nine-Point Circle
2.5 The Simson Line
2.6 Generalizations of the Simson Theorem
2.7 The Cantor Theorems
2.8 The Feuerbach Theorem
2.9 The Morley Theorem
Exercises

3 Modius Transformations
3.1 Stereogrpahic Projection
3.2 Mobius Transformations
3.3 Cross Ratios
3.4 The Symmetry Principle
3.5 A Pair of Circles
3.6 Pencils of Circles
3.7 Fixed Points and the Classification of Mobius Transformations
3.8 Inversions
3.9 The Poincare Model of a Non-Euclidean Geometry
Exercises

Epilogue

A Preliminaries in Geometry
A.1 Centers of a Triangle
A.1.1 The Centroid
A.1.2 The Circumcenter
A.1.3 The Orthocenter
A.1.4 The Incenter and the Three Excenters
A.1.5 Theorems of Ceva and Menelaus
A.2 Angles Subtended by an Arc
A.3 The Napoleon Theorem
A.4 The Apollonius Circle

B New Year Puzzles
Index