Let's assume that none of the triangles share any vertices, then 12 points would make 4 triangles.
It takes two points to form a base, so 6 points would form the bases of 3 triangles, and these bases lie on the same line, so we would need a 4th point in line with the other. This could be the vertex of the 4th triangle.
For example:
The 7 collinear points are A, B, D, E, G, H, L, The remaining points (vertices) C, F, I, J, K are not collinear.