I think 6 lines of 3 points each is possible, with 4 points, tricky.
4 points x 6 lines = 24 points if not sharing any points
if we only have 10 points, that means 14 points has to be shared with x number of 10 points,
1) can do 7 points sharing among 3 lines, and 3 points not sharing with any line.
2) can do 4 points sharing among 4 lines, 2 points sharing between 2 lines, and 4 points not sharing with any line.