Advanced math and axiomatic euclidean geometry

Are you so busy you can't invest 3 days with a book and see where it takes you?

I think it helps

If you are going to do anything geometric, axiomatic geometry is very useful: not only the basics (parallelism, orthogonality, all that), but also the more sophisticated things (projective geometry, inversions, etc.).
It helps you build a solid understanding of group theory grounded in concrete stuff, as well as non-euclidean geometry and projective spaces.
Now, of course, you cannot skip coordinate geometry, linear algebra, commutative algebra, etc. (as it is, indeed, the basis for *algebraic* geometry) but not everything in coordinate-free geometry is to be thrown out, quite the contrary.

>>7963681
The thing is, I understand projective geometry, parallelism and orthogonality through linear algebra.

To me, projective space is a union of affine hyperspaces, parallel lines are those satisfying a system of linear equations with no solutions, orthogonality is value of bilinear function on vectors, inversion is just a map.

That's why building trivial stuff, like orthogonal lines in hyperbolic geometry is pain to me: I'm not used to thinking with synthetic geometry.

>>7963723
I agree that many things about projective spaces can be understood in terms of linear algebra and it is in many situations the easiest and cleanest way to prove things about projective spaces. However, when you are *investigating* things (ie. when you don't know what you want yet), any intuition can be useful, and working with synthetic geometry gives you a lot of intuition and helps you develop good reflexes (drawing the situation and seeing where you can get from there, choosing an appropriate hyperplane at infinity, using transformations to reduce a problem to an easier one).

For hyperbolic spaces, it might be because your differential geometry background is lacking.
Basically, if you can define an inner product "at every point", then you can define the angle between two curves as the angle between their tangent vectors. Now, differential geometry (more specifically riemannian geometry) provides a framework for talking about these sort of things (with vector bundles and all that).
Also, working with differential geometry will help you see why inversion is more than "just a map" and actually a very useful transformation.

>>7963772
Talking about hyperbolic spaces, we had to contruct perpendicular using only inversions.

The part about intuition is reasonable...