A line is defined by its slope and a point. For every point i, we calculate the slope to every other point j. We use a hash map to count the number of points for each slope.

1. For each point p1 in points:
   • Initialize slopes = {}.
   • For every other point p2:
     • Calculate slope (y2 - y1) / (x2 - x1).
     • To avoid floating point issues, use GCD: dy / gcd, dx / gcd as the key.
     • Special case for vertical lines.
     • Update slopes[slope].
   • Track the maximum count for point p1.
2. Return global maximum.

• Time Complexity: O(N^2).
• Space Complexity: O(N).

import math

def max_points(points):
    n = len(points)
    if n <= 2: return n
    
    res = 1
    for i in range(n):
        slopes = {}
        for j in range(i + 1, n):
            dx = points[j][0] - points[i][0]
            dy = points[j][1] - points[i][1]
            g = math.gcd(dx, dy)
            slope = (dx // g, dy // g)
            slopes[slope] = slopes.get(slope, 0) + 1
            res = max(res, slopes[slope] + 1)
            
    return res