Here are links to some recent generalizations of the Gallai-Sylvester theorem.
B. Barak, Z. Dvir, A. Wigderson, A. Yehudayoff Fractional Sylvester-Gallai theorems, Proceedings of the National Academy of Sciences of the United States of America 2012. (Link to a journal proceeding.)
A. Ai, Z. Dvir, S. Saraf, A. Wigderson Sylvester-Gallai Type Theorems for Approximate Collinearity Forum of Mathematics, Sigma, vol. 2, 2014.
See also this blog post on fractional Gallai-Sylvester.