Books on classical geometry

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 ^^

Solution 1:

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.)

Solution 2:

For a careful and very comprehensive treatment:

Robin Hartshorne: Geometry: Euclid and Beyond; Springer, 2000.