<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>

class Solution:
    def commonDivisor(self, x: int, y: int) -> int:
        if x < 0:
            x = -x
        if y < 0:
            y = -y
        if x < y:
            x, y = y, x
        while y > 0:
            x, y = y, x % y
        return x

    def maxPoints(self, points: List[List[int]]) -> int:

        # (x-x0)(y1-y0) == (x1-x0)(y-y0)
        # x y1 + x0 y0 - x0 y1 - x y0 == x1 y + x0 y0 - x1 y0 - x0 y
        # x (y1 - y0) - x0 (y1 - y0) == y (x1 - x0) - y0 (x1 - x0)
        # Hash key = (x1 - x0, y1 - y0, x0 y1 - x1 y0), value = indices
        table = {}
        for i in range(len(points)):
            for j in range(i + 1, len(points)):
                x0, y0 = points[i]
                x1, y1 = points[j]
                a = x1 - x0
                b = y1 - y0
                c = x0 * y1 - x1 * y0
                if a < 0 or (a == 0 and b < 0):
                    a, b, c = -a, -b, -c
                d = self.commonDivisor(a, b)
                a, b, c = a / d, b / d, c / d
                s = table.get((a, b, c))
                if not s:
                    s = set()
                    table[(a, b, c)] = s
                s.add(i)
                s.add(j)
        n = 1 if points else 0
        for _, indices in table.items():
            n = max(n, len(indices))
        return n
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