<p>Given an array of <code>points</code> where <code>points[i] = [x<sub>i</sub>, y<sub>i</sub>]</code> represents a point on the <strong>X-Y</strong> plane, return <em>the maximum number of points that lie on the same straight line</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/02/25/plane1.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> points = [[1,1],[2,2],[3,3]]
<strong>Output:</strong> 3
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/02/25/plane2.jpg" style="width: 300px; height: 294px;" />
<pre>
<strong>Input:</strong> points = [[1,1],[3,2],[5,3],[4,1],[2,3],[1,4]]
<strong>Output:</strong> 4
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= points.length &lt;= 300</code></li>
	<li><code>points[i].length == 2</code></li>
	<li><code>-10<sup>4</sup> &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 10<sup>4</sup></code></li>
	<li>All the <code>points</code> are <strong>unique</strong>.</li>
</ul>

from collections import defaultdict
class Solution:
    def maxPoints(self, points: List[List[int]]) -> int:
        L = defaultdict(set)
        def line(points, i, j):
            x0, y0 = points[i]
            x1, y1 = points[j]
            a , b = y1 - y0, x0 - x1
            c = x0 * a + y0 * b
            d = gcd(gcd(a, b), c)
            a, b, c = a // d, b // d, c // d
            if c < 0:
                a, b, c=-a, -b, -c
            elif c==0:# This can occur!
                if b<0 or (a<0 and b==0): 
                    a=-a 
                    b=-b
            line_key = (a, b, c)
            L[line_key].add(i)
            L[line_key].add(j)

        n = len(points)
        if n == 1:
            return 1  # edge case

        for i in range(n - 1):
            for j in range(i + 1, n):
                line(points, i, j)

        max_points_on_line = 0

        for _, point_set in L.items():
            max_points_on_line = max(max_points_on_line, len(point_set))

        return max_points_on_line
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