Question: What is Geometry Revisited by Coxeter?

[micromass]

• Author: H. S. M. Coxeter, Samuel L. Greitzer
• Title: Geometry Revisited
• Amazon Link: https://www.amazon.com/dp/0883856190/?tag=pfamazon01-20
• Prerequisities: High-School mathematics
• Level: Undergrad

Table of Contents:

[LIST]
[*] Preface
[*] Points and Lines Connected with a Triangle
[LIST]
[*] The extended Law of Sines
[*] Ceva's theorem
[*] Points of interest
[*] The incircle and excircles
[*] The Steiner-Lehmus theorem
[*] The orthic triangle
[*] The medial triangle and Euler line
[*] The nine-point circle
[*] Pedal triangles
[/LIST]
[*] Some Properties of Circles
[LIST]
[*] The power of a point with respect to a circle
[*] The radical axis of two circles
[*] Coaxal circles
[*] More on the altitudes and orthocenter of a triangle
[*] Simson lines
[*] Ptolemy's theorem and its extension
[*] More on Simson lines
[*] The Butterfly
[*] Morley's theorem
[/LIST]
[*] Collinearity and Concurrence
[LIST]
[*] Quadrangles; Varignon's theorem
[*] Cyclic quadrangles; Brahmagupta's formula
[*] Napoleon triangles
[*] Menelaus's theorem
[*] Pappus's theorem
[*] Perspective triangles; Desargues's theorem
[*] Hexagons
[*] Pascal's theorem
[*] Brianchon's theorem
[/LIST]
[*] Transformations
[LIST]
[*] Translation
[*] Rotation
[*] Half-turn
[*] Reflection
[*] Fagnano's problem
[*] The three jug problem
[*] Dilatation
[*] Spiral similarity
[*] A genealogy of transformations
[/LIST]
[*] An Introduction to Inversive Geometry
[LIST]
[*] Separation
[*] Cross ratio
[*] Inversion
[*] The inversive plane
[*] Orthogonality
[*] Feuerbach's theorem
[*] Coaxal circles
[*] Inversive distance
[*] Hyperbolic functions
[/LIST]
[*] An Introduction to Projective Geometry
[LIST]
[*] Reciprocation
[*] The polar circle of a triangle 
[*] Conics
[*] Focus and directrix
[*] The projective plane
[*] Central conics
[*] Stereographic and gnomonic projection
[/LIST]
[*] Hints and Answers to Exercises
[*] References
[*] Glossary
[*] Index 
[/LIST]

 

[BloodyFrozen]

Good for high school math competitions. Almost all the topics are not covered in a standard high school math course.

 

[Snow-Leopard]

The Book is used for AMC VIII to AMC XII.The best book ever written for Mathematics Olympiad Geometry.

 

[Vargo]

It seems that the best thing people can say about this book is that it helps you to win high school math competitions.

I read this book after finishing my undergraduate degree in mathematics. I found it enjoyable, but I preferred Coxeter's Introduction to Geometry because it had more depth and breadth.

 

[mathwonk]

I agree. I am not nuts about this book. Winning contests involves using facts that you may not understand fully how to prove. This book is like that. E.g. the discussion of the "power of the point" claims correctly that this theorem of Euclid is an easy corollary of the principle of similarity. True enough.

However what they do not mention is that the theory of similarity is quite deep, and was not available to Euclid when he proved this theorem, so he gave a different proof using Pythagoras. Indeed if one uses Euclid's proof, then one can use this result to deduce the important principle of similarity without going to as much difficulty as is usually done.

If like me you are interested in the logical connections between different results, then you believe in doing them in logical order, not assuming the most difficult and deep ones first without justification, and then using them to make other results appear easy.

If however you want to solve contest problems quickly, then you want to use all the big guns available on the littlest peanut problems, in order to dispatch them in enough time to finish the test with the highest possible score. There is no harm in this, and I was myself so motivated in high school, but not so much any more.

 

[fourier jr]

the only thing I remember about this book is it's the place where I found out how to solve the 3 jugs problem using barycentric coordinates:
http://www.cut-the-knot.org/triangle/glasses.shtml