Research in plane geometry or euclidean geometry I was doing good at school in plane geometry and trigonometry - especially in geometric proofs like proving the equality of two line segments or two angles - more than I was doing in analytic geometry.
I am considering doing research in mathematics to be my career (and my life) someday. and I am wondering about the most interesting research area for me.
What I am asking about is :

Is there still open research fields in plane geometry or Euclidean geometry? Or this area is considered to be fundamentals that are already investigated enough?

Thank you!
 A: You can find many papers about recent work in advanced "classical" Euclidean geometry at this site:  http://forumgeom.fau.edu/ 
A: I am not sure this is something you are looking for. But I find it interesting: it is easy to understand and is kind of related to plane geometry: 
A graph is planar if it can be drawn on the plane without any crossing. It is well known that every planar graph can be drawn on the plane such that every edge is drawn as a straight line segment. Kemnitz and Harborth made the following stronger conjecture:
Conjecture. Every planar graph can be drawn on the plane such that every edge is a straight line segment with integer edge lengths.
For the case when the graph is cubic (that is, every vertex has exactly three edges connecting to it), it was known to be true. See journal link or paper. But the general case is still open. You can understand the problem more by reading the paper. 
A: Actually there is a theorem of Tarski that elementary Euclidean geometry is decidable. Roughly speaking this means that there is a computer program that can decide if a given statement of elementary Euclidean geometry is true (given enough time). Generally mathematicians do not like to do things that can be done on a computer. And so there is no active research in elementary geometry. There is some information on this in wikipedia http://en.wikipedia.org/wiki/Euclidean_geometry#Logical_basis. You can also check this question out Is it possible to solve any Euclidean geometry problem using a computer?.
