Does anyone know of a proof of the fact that any 2manifold can be triangulated that does not use the JordanCurve Theorem or the JordanSchoenflies Theorem? Thanks for your help

Take the proof that any compact smooth manifold admits triangulations, and set the dimension to two. The idea goes like this:
Paraphrasing Allen Hatcher: If you're interested in topological surfaces, the paper A.J.S. Hamilton, The triangulation of 3manifolds, Oxford Quart. J. Math. 27 (1976), 6370 takes the KirbySiebenmann machinery and scales it down to 3 dimensions where it becomes somewhat simpler, so one can prove existence and uniqueness of triangulations of 3manifolds using only standard PL techniques, such as results of Waldhausen. Presumably the same approach would work for surfaces. Since the method works in 3 dimensions it can't be using the topological Shoenflies theorem since this fails in 3 dimensions. On the other hand, it would use some PL (or smooth) surface theory so it wouldn't be entirely "from scratch". edit: Allen wrote this argument up in a recent paper. See this thread for details http://mathoverflow.net/a/151760/353 

