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I am interested in reading Coxeter's famous text Geometry Revisited. It's not clear to me what the prerequisites for this text are, however. I'm sure I have enough mathematical maturity: I know Analysis at the level of Baby Rudin, a fair chunk of the group theory and ring theory from Dummit and Foote, and some elementary number theory (quadratic reciprocity, basic theorems on UFD, etc.). My problem is that when it comes to even basic geometry, I am incredibly ignorant (indeed, this is why I want to read Coxeter's text). For example, when reading an online preview of the book, Coxeter cites without reference or proof that any triangle is cyclic; I realize this is a very fundamental theorem, but it is one I did not know of and would like to see a proof of. I am sure there are plenty of other other very basic and well-known results about triangles, quadrilaterals, circles, and so forth, that I do not know, and that may be used in the book without proof.

Is it still possible to work through the text? My fear because I have so little background, will not be able to complete the exercises in the text, since they will likely depend on using basic facts not mentioned in the text. So, if it's not feasible for me to work through the book, where can I acquire the necessary fundamental knowledge? Again, I have some mathematical maturity, so a text that introduces the necessary concepts while sticking to the "definition-theorem-proof" style of rigorous mathematics is ideal. Does such a book exist, or is it best to simply dive into Coxeter, and try to pick up the relevant facts as I proceed through it?

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  • $\begingroup$ Euclid is always a good choice...but you could also just start reading Coxeter (great book, by the way) and search an on line copy of Euclid (there are several freely available) for results you aren't familiar with. $\endgroup$ – rogerl Oct 23 '14 at 18:35
  • $\begingroup$ The Art and Craft of Problem Solving (second edition) by Paul Zeits has a chapter called "Geometry for Americans" that might be interesting. $\endgroup$ – littleO Jan 28 '15 at 7:16
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A classic is Hadamard's Lessons in Geometry, recently translated into English. The original French version is free on the internet (legally).

If you do choose to read Euclid directly, as proposed in a comment, you might want to supplement your reading with Hartshorne's Geometry: Euclid and Beyond to get a modern perspective.

I think Pogorelov's Geometry might (partially) fit your needs. (Note: This Mir book is not a translation of his high school textbook on geometry. Its intended audience is future high school teachers. His high school book has been translated into Spanish, but not English.) It begins with analytic geometry but also has discussion of a rigorous axiom system for Euclidean geometry in a later part, and some construction problems in elementary geometry, as well as a small amount of differential geometry.

Finally, I'd like to mention that although Geometry Revisited is a terrific book, it also seems to consciously avoid using coordinates, even in cases where this would simplify an argument considerably. Dan Pedoe's Geometry: A Comprehensive Course may strike a better balance on some of the same material.

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