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Since you didn't mention memory, I'll assume O(N^2) is acceptable.

Setup pseudocode:
```
for every p1, p2:
    ln = line(p1, p2)
    points_on_line[ln.m, ln.c] += [p1, p2]
```

If there are at least 2 colinear points on your line, you can get them in O(1) with `points_on_line[m, c]`. If not, you'll have to do brute force.

The initialization takes O(N^2) so it is practical if the number of searches you need to do is much more than `N^2` (eg. 5 points, 1000 searches). Note that you can add points:
```
def add_point(p):
    for m, c in points_on_line:
        if p.y == m * p.x + c:
            points_on_line[m, c] += [p]
```

This will be O(N), so adding new points is okay so long as you are doing a lot of searches for each point you add.