wishing to learn more geometry

[khemix]

if you did plane / euclidean geometry in highschool, what is next?

ive looked into non-euclidean geometry, but it seems to use weird algebra i have never seen. is the classical geometry of constructions finished as of now, and will all geometry from now on be very algebraic?

there is no courses in geometry in my school and there is little information about it. so i would like to know to learn.

[Tac-Tics]

What parts of geometry do you like?

As far as I know, there's no obvious "step up" from high school geometry. There are many other subjects that are heavily founded in geometry, so it might help to know what you really enjoyed about it.

[khemix]

i dont know what parts i like, i only did plane geometry, and i really enjoyed it - the different properties of triangles and stuff was very interesting.

im not interested in differential geoemtry or topology and the like at the moment, i just want to continue doing the classical geometries (ie. things that dont use calculus). i really dont know what is studied beyond plane geometry, which is why im not even sure what it is i am looking for. i heard its natural to go to non-euclidean geometry after plane geometry, so my guess is that is what is next.

[robphy]

Trigonometry and Analytic Geometry
Vector Algebra applied to Geometry
Solid Geometry
Projective Geometry
Minkowskian geometry (for Special Relativity)

Here's a neat site: http://www.cut-the-knot.org/geometry.shtml

[yenchin]

How much have you learned in classical plane geometry? Have you done things like Menelaus and Ceva theorem? There are a lot of pretty stuffs in geometry that are not covered in high school syllabus, but they appear frequently in mathematical olympiads.

You may want to read, if you have not already, Coxeter's book "Geometry Revisited (w.amazon.com/Geometry-Revisited-Mathematical-Association-Textbooks/dp/0883856190)"

If you know, or don't mind to learn, some complex numbers, you can do many interesting geometry problems using complex numbers. I recommend "Complex Numbers and Geometry" (http://www.amazon.com/Complex-Geometry-Mathematical-Association-Textbooks/dp/0883855100).

[mathwonk]

euclid's own book is a big step up from high school geometry, or hartshorne's book, geometry, euclid and beyond.

http://www.amazon.com/Euclids-Elements-T-L-Heath-Translation/dp/1888009195/ref=sr_1_1?ie=UTF8&s=books&qid=1226627334&sr=1-1


http://www.amazon.com/Geometry-Euclid-Beyond-Robin-Hartshorne/dp/0387986502/ref=sr_1_4?ie=UTF8&s=books&qid=1226627396&sr=1-4


in fact in most high schools, even a basic book like geometry by harold jacobs is much better. (the second edition is preferred to the third.)
http://www.amazon.com/Geometry-Harold-R-Jacobs/dp/071671745X/ref=sr_1_2?ie=UTF8&s=books&qid=1226627552&sr=1-2