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?
Relevance of Euclidean geometry to modern mathematics
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soft-question
euclidean-geometry
education
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1From the point of view of modern mathematics, see http://en.wikipedia.org/wiki/Erlangen_program . – 2012-09-08