Characterizing triangles unembeddedly The mathedu mailing list has a recent longish thread at
http://www.nabble.com/Why-do-we-do-proofs--to25809591.html
which discussed among other things whether we should teach triangles as labeled or unlabeled to high school students (this is a vast oversimplification of the thread).  I have long been concerned with how we think (informally and formally) about mathematical objects, so naturally I started to consider how we think about triangles.
Consider circles.   Most informal and formal descriptions involve an embedding into R^2, but they can be characterized as manifolds (even as Riemannian manifolds) of dimension 1 with specific properties, independent of any embedding. This sort of thing has turned out to be a major way to think about all sorts of spaces.  So can we describe triangles in a similar way?
Unfortunately, manifolds are far removed from my usual mathematical work (category theory).  What I think I understand is that there can be piecewise linear manifolds, even Riemannian ones.  So perhaps we can say a triangle is a piecewise linear manifold of dimension 1 with certain properties.  Now, I want to define a triangle so that it comes complete with information about the lengths of its sides and what the three angles are.  Riemannian manifolds have a way to specify length and angles, and I can believe you can make the sides have specific lengths.  But the angles?  It seems to me that the tangent spaces (like those on a circle) result in all angles being 0 or pi, except at the corners where they don't exist.  But I may not understand the situation correctly.
So my question is:  Is there a known methodology that allows triangles to be characterized independent of embeddings in such a way that incorporates information about side lengths and angles?
 A: I'm not sure what you have in mind for circles--the only invariant of a connected compact Riemannian 1-manifold is its length; they have only extrinsic curvature (defined for a manifold embedded in Euclidean space) not intrinsic curvature. You'd have to consider Riemannian manifolds with some extra structure--perhaps just a function which is supposed to be the curvature of an embedding into R^2, and integrates to 2π.  Then you could represent triangles similarly, with a curvature "function" which is the sum of three delta functions, one at each vertex.  Of course the side lengths of a triangle determine its angles, so you might produce a "triangle" which cannot be embedded in R^2!
Interestingly for the case of 2-manifolds the situation is nicer.  There is a theorem that a Riemannian 2-manifold of positive curvature homeomorphic to the sphere has a unique (up to rigid motion) isometric immersion in R^3 as a convex surface.  There is an analogous statement for polyhedra but I am not sure of the precise statement.  I think these results are due to Pogorelov and Alexandrov.
A: I think the answer depends on what you want to do with your triangles.  If you want to talk about abstract triangles given by lengths and angles, you can characterize them by a collection of 6 numbers, 3 of which give side lengths (presumably satisfying a triangle inequality), and 3 of which specify angles between 0 and pi.  There are then some conditions (e.g., law of sines) that need to be satisfied if you want an embedding of such a triangle in Euclidean space, or more generally, a constant curvature surface, such that sides are geodesics and angles are preserved.  It's not clear to me that this abstract framework is useful.
