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## Geometry Revisited by Coxeter

Code:
 Preface
Points and Lines Connected with a Triangle 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

Some Properties of Circles 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

Collinearity and Concurrence Quadrangles; Varignon's theorem
Napoleon triangles
Menelaus's theorem
Pappus's theorem
Perspective triangles; Desargues's theorem
Hexagons
Pascal's theorem
Brianchon's theorem

Transformations Translation
Rotation
Half-turn
Reflection
Fagnano's problem
The three jug problem
Dilatation
Spiral similarity
A genealogy of transformations

An Introduction to Inversive Geometry Separation
Cross ratio
Inversion
The inversive plane
Orthogonality
Feuerbach's theorem
Coaxal circles
Inversive distance
Hyperbolic functions

An Introduction to Projective Geometry Reciprocation
The polar circle of a triangle
Conics
Focus and directrix
The projective plane
Central conics
Stereographic and gnomonic projection

References
Glossary
Index

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 Recognitions: Gold Member Good for high school math competitions. Almost all the topics are not covered in a standard high school math course.
 The Book is used for AMC VIII to AMC XII.The best book ever written for Mathematics Olympiad Geometry.

## Geometry Revisited by Coxeter

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.
 Recognitions: Homework Help Science Advisor 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.
 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

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