I’m curious to whether you guys have any tips on book concerning classical euclidean geometry. I’d like somewhat of an advanced treatment, around the same level as Coxeter’s “Geometry revisited”. I’d however want something more comprehensive, therefore I’m asking you for tips!
Hopeful for answers ^^
A book I like very much is the classic “College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle” by Nathan Altshiller-Court. The (revised in 1952) book has been reissued recently (2007) by Dover in a very affordable paperback. There is also a companion book on spatial geometry, but I do not know how easy it would be to find a copy. In Colombia, training for the Math Olympiads used for a while this book and Coxeter’s, so I have first hand experience of learning from it.
For completely different reasons, another book I like a lot is “Higher Geometry” by N.V. Efimov, originally from Mir eds. The book begins with an interesting discussion of axiomatic systems, talks about the attempts to prove the 5th postulate, Hilbert’s axiomatization of Euclidean geometry, and then develops it synthetically. The book then does the same with hyperbolic geometry, and this is the only place where I have seen this done (it is well worth reading this part). There is also a treatment of projective geometry which is very nice, and (I think) of differential geometry. (The description in this book of the foundational approach was one of the reasons I decided I wanted to specialize in logic.)
For a careful and very comprehensive treatment:
Robin Hartshorne: Geometry: Euclid and Beyond; Springer, 2000.
Well I only took one course on elementary geometry so I don’t know much about this, but we used Edwin Moise’s book Elementary Geometry From an Advanced Standpoint. It is very well written and probably has a level that will suit you. We used the first edition so I’m not sure what may have changed in the other editions and I can’t seem to find a google preview of the third edition.
In any case I highly recommend you take a look at it to see if you like it.
Edit
I found the table of contents for the third edition here.
Hartshorne and Coxeter’s books are very good. If you like challenging problems, you may also enjoy Kedlaya’s Geometry Unbound. At least 10 years ago, this is the book the US Olympiad team learned geometry from.
How about:
The Foundations of Geometry by David Hilbert
The Four Pillars OF Geometry by John Stillwell.
The link contains the book. Hopefully you shall enjoy reading it.
Try Elementary Geometry by Ilka Agricola and Thomas Friedrich. It has nice expositions of two-dimensional spherical and hyperbolic geometries as well.
Christopher Gibson’s Elementary Euclidean Geometry is a new (published in 2004) and fairly advanced introduction to the geometry of lines and conics in Euclidean space.
What is you objective? It is difficult to make sensible suggestions without knowing that. For example, if you want to get better at olympiads or similar more advanced contests, then I strongly recommend problems books (or just problems), not text books.
If you are really interested in C1 differential topology, or manifolds, or elliptic curves or something, then the answers will be totally different. You asked about classical Euclidean, but the trouble is that is the jumping off point for all kinds of subjects.
For a complete beginner,
Kiselev’s Geometry.
Now read the “What is Geometry” part of Geometric Transformation 1 by Yaglom. You can find the book in “scribd” as pdf.
The great “Geometry Revisited” + Geometry Unbound. // You may first read “Plane Euclidean Geometry: Theory and Problems” by UK Math Trust. It is highly recomended and best for medium level geometry. It is intended for Olympiads.
Problems in Plane Geometry by Parsolov.
Now take “Geometry: Euclid and Beyond” or go with the classic and a superb book by David Hilbert, Geometry and the Imagination.
All the books except Plane Euclidean Geometry can be found on net.search in the scribd.com
Here is one at advanced high school level: Plain Plane Geometry by Amol Sasane: http://www.worldscientific.com/worldscibooks/10.1142/9907