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I'm interested whether, barring the reasons such as the importance of historical narrative and having an illustrative example, one should learn (synthetic) Euclidean geometry.
To make such an inquiry a bit less subjective, I'm interested as to what is the relevance of Euclidean geometry to modern mathematics. From a point of view of modern mathematics, is there any 'deep' mathematics in Euclidean geometry? Even if there are some important lessons to be learned from Euclidean geometry, is it possible to present them in a more purified form, rather than learning the whole of classical Euclidean geometry?

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From the point of view of modern mathematics, see en.wikipedia.org/wiki/Erlangen_program . –  Qiaochu Yuan Sep 8 '12 at 19:52

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I would take "the whole of classical Euclidean geometry" to mean every result in the Elements that we would now consider to be geometry, as opposed to, say, number theory, algebra, or the geometric series.

Are there "deep" results? I suppose it depends on what you mean by deep. Ancient Greeks would probably have perceived it as deep that the Pythagorean theorem could be proved from the postulates. In a modern context, everything in Euclid is material that is (or could be) taught to high school students, so nobody would consider it deep. However, there are some pretty deep results about Euclidean geometry that aren't in the Elements, e.g., Banach-Tarski.

The tone of the question suggests that the OP finds it tedious to wade through proof after proof involving triangle ABC and parallelogram PQRS, and expects that studends will also find this tedious. This is a matter of taste, style, and presentation. If one wishes, one can do all of Euclid from a Cartesian perspective. Certainly nobody today would bother using Archimedes' methods to prove Archimedes' classic result about the volume of a sphere. We would use calculus.

Coxeter's book Introduction to Geometry may be helpful to the OP. This is not a high school text but rather a book aimed at upper-division college math majors. It's very entertaining and interesting, not bogged down with page after page of parallelogram PQRS. It also puts Euclidean geometry in the context of other geometries, and applies modern techniques such as conformal transformations. Euclidean geometry lies at the core of all of modern geometry. It's the reference point for other systems, e.g., a manifold is a space that's locally Euclidean (in the sense of homeomorphism).

If the question is whether or not a student like my 13-year-old daughter should be taking geometry this year in school, then I think the answer should be obvious. She needs to learn the Pythagorean theorem, for example, and to be exposed to formal proof for the first time. A completely different question is whether the subject should be presented in a particular style or whether particular topics should be included. Certainly it would be silly to include every geometrical result in the Elements, or to eschew modern tools such as the real number system and the notion of measuring angles greater than 180 degrees. It would also be silly to make her learn geometry from a book that is all Cartesian and has no diagrams.

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Thanks for the answer Ben. I'm not sure if I agree that Banach-Tarski is a result about Euclidean geometry. Sure, the setting is Euclidean space, but the things about which it talks are miles away from discussions of similarity of triangles or congruence of angles. –  user5501 Sep 8 '12 at 15:28
    
I'll check out the Coxeter's book. While you have, in a certain sense, read correctly the tone of my question, I also have some amount of 'respect' for Euclidean geometry, I have read parts of the Elements and I'm not sorry for the time I have spent learning Euclidean geometry -- I posted this question to find on which counts I was wrong, not as a vehicle to bash Euclidean geometry. –  user5501 Sep 8 '12 at 15:31
    
@LovrePešut: Banach-Tarski is manifestly a result about Euclidean geometry. But one could argue that it's in some sense a result demonstrating what is wrong with a certain embodiment of Euclidean geometry. That is, if you treat Euclidean geometry using point sets and ZFC, you get results like Banach-Tarski that we would like to be false in a system designed to describe our intuitions about physical space. In that sense, Banach-Tarski is really a result about the axiom of choice, and why we should not necessarily accept AC as "true." –  Ben Crowell Sep 8 '12 at 17:19

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