15 points

If we want to choose three points, we can do it $\binom{15}{3}=\dfrac{15*14*13}{1*2*3}=455$ ways. The three points always define a triangle, except they are on the same line. There are three lines, and there are $6$ points in each of them. Since three points can't be on two lines at the same moment, from the $455$ ways $3*\binom{6}{3}=3*\dfrac{6*5*4}{1*2*3}=60$ ways aren't make solutions.

Therefore the answer is $455-60=\boxed{395}$ .