35 triangles found when I quit ignoring some lines in a misguided optimization. Thanks again to Perl Mouse for posting something which showed that I was missing some results.
27 triangles found when I accounted for lines that don't actually exist. This corrects the problem that Perl Mouse found.
46 triangles found when I stopped to remove lines. Again, this takes no time worth reporting.
51 triangles found, no runtime worth reporting. Except that drat, I didn't notice that some "triangles" are actually just line segments.