The cubic curve $2x^3-4x^2y+2xy^2-8x+y^3-y$ can be used to get lattice points allowing the placement of the numbers $-8$ to $8$ so that all 32 triplets that sum to 0 will be a straight line of three. Some other nice curves are at "Elliptic Curves on a Small Lattice"
Is there such a diagram for -9 to 9?
EDIT -- Lines through 1-8, 2-7, 3-6, 4-5 turn out to be concurrent. So it turns out to be easy to add 9 and -9. Similarly, -10/10 and -11/11 can be added. -12/12 also seems concurrent, but it's a far off point. Points for -13/13, -14/14, and -15/15 are also on concurrent lines.
So I give a solution, that seemingly extends to infinity, but I don't have an explanation. Is there a less crowded diagram for -14 to 14? Are these easy to make by some method? Can more digits be added to infinity with all triples concurrent?
EDIT 2: $(x,y)$ positions for 0 to 17 are as follows, with the third item being the denominator:
$(0,0,1), (1,-3,1), (0,-1,1), (1,-1,1), (-1,-2,1), (3,1,1), (-4,-3,1),$
$(3,3,1), (-2,0,1), (3,11,7), (-4,7,5), (-1,13,19), (-3,10,1), (1,-9,23),$
$(24,-57,31), (-11,-87,67), (84,-32,59), (-143,-187,73)$
For -1 to -17, take the negative. Is there an easy way to get these values from the original curve?