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I want to self-study geometry for interest. What I want to study is closed to the fundamental appeal, not problem-solving, not differential geometry. But I found that there were many of books on geometry. And it is difficult for a beginner like me to figure out which books are the ones I need.

I want to learn the axiomatic/synthetic, ancient/modern fundamentals of geometry, such like what is "point", "segment", "directed segment", "Euclidean vector", "plane", "congruence", "affine space", etc; and what is the modern way of defining and treating these concepts? If there's a book compare those difference is also what I want. I guess this kind of books should be of graduate level right? I've studied analysis and linear algebra, so a book uses plenty of terminology of these subject is OK for me!

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marked as duplicate by rschwieb, I am Back, mrp, Leucippus, hardmath Apr 12 '17 at 4:46

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    $\begingroup$ I would recommend Hartshorne's Euclid and beyond for a modern treatment with some context of the past. $\endgroup$ – rschwieb Apr 10 '17 at 13:10
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Here are two texts on Geometry which base the treatment on Linear Algebra:

Audin -- Geometry (2003)

Gruenberg & Weir -- Linear Geometry, 2nd Ed (1977)

Here's one that does Geometry based on a combination of analytic tools$\,-\,$coordinates, Calculus, as well as lots of Linear Algebra:

Brannan & others -- Geometry, 2nd Ed (2012)

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  • $\begingroup$ Thanks. Did Audin's Geometry briefly mention why he adopt the linear algebra approach but not the others? Or did he explain at first or at end the benefit of such approach? $\endgroup$ – Eric Apr 10 '17 at 13:39
  • $\begingroup$ From Audin's introduction: "This is a book written for students who have been taught a small amount of geometry at secondary school and some linear algebra at university. ... Two directing ideas. The first idea is to give a rigorous exposition, based on the definition of an affine space via linear algebra, but not hesitating to be elementary and down-to-earth. I have tried both to explain that linear algebra is useful for elementary geometry ... and to show “genuine” geometry: triangles, spheres, polyhedra, angles at the circumference, inversions, parabolas. . ." $\endgroup$ – quasi Apr 10 '17 at 14:03

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