<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 maxPoints(self, points: List[List[int]]) -> int:
        # if on a line means on y = kx + b, y2 - y1/x2 - x1, (y2 - y1)x - (x2 - x1)y + x2y1 - x1y2 = 0
        # how to avoid x3, y3 x2, y2? visited set add(k, b)
        n = len(points)
        if n <= 2:
            return n

        visited = set()
        ans = 2
        for i in range(n):
            for j in range(i + 1, n):
                x1, y1 = points[i]
                x2, y2 = points[j]
                if x1 != x2 and ((y2 - y1) / (x2 - x1), y1 - ((y2 - y1) / (x2 - x1)) * x1) in visited:
                    continue
                if x1 == x2 and (float('inf'), x1) in visited:
                    continue
                ans = max(ans, self.countPoints(points, visited, x1, y1, x2, y2))
        return ans

        

    def countPoints(self, points, visited, x1, y1, x2, y2):
        ans = 2
        for x, y in points:
            if (x == x1 and y == y1) or (x == x2 and y == y2):
                continue
            if (y2 - y1) * x - (x2 - x1) * y + x2 * y1 - x1 * y2 == 0:
                ans += 1
        visited.add(((y2 - y1) / (x2 - x1), y1 - ((y2 - y1) / (x2 - x1)) * x1) if x1 != x2 else (float('inf'), x1)) 
        return ans

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