Among the many beautiful and nontrivial theorems in geometry found here are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. A nice proof is given of Morley’s remarkable theorem on angle trisectors. The transformational point of view is emphasized: reflections, rotations, translations, similarities, inversions, and affine and projective transformations. Many fascinating properties of circles, triangles, quadrilaterals, and conics are developed.
1. Points and Lines Connected with a Triangle
2. Some Properties of Circles
3. Collinearity and Concurrence
5. An Introduction to Inversive Geometry
6. An Introduction to Projective Geometry
7. The Calculus of Variations and Geometry
8. A Glimpse at Higher Dimensions
Hints and Answers to Exercises
H.S.M. Coxeter (February 9, 1907-March 31, 2003) is regarded as one of the greatest geometers of the 20th century. He was most noted for his work on regular polytopes and higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra.
S.L. Greitzer (August 10, 1905-February 22, 1988) was the founding chairman of the committee in charge of the United States of America Mathematical Olympiad, and the publisher of the pre-college mathematics journal Arbelos. His work with the USAMO and the IMO made him well known in competition circles world-wide.
In 1961 a book appeared with the widely embracing title Introduction to Geometry. Its author was H. S. M. Coxeter who, in the preface, said that ‘For the past thirty or forty years, most Americans have somehow lost interest in geometry. The present book constitutes an attempt to revitalize this sadly neglected subject’. Continued...