<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:
        # each edge can be mapped to a pair (a, b)
        # where a=slope, b=bias of the unique line containing edge
        from collections import defaultdict
        import fractions
        from itertools import combinations
        
        lines = defaultdict(int)
        vert_lines = defaultdict(int)
        
        for p1, p2 in combinations(points, 2):
            x1, y1 = p1
            x2, y2 = p2

            if x1 == x2:
                vert_lines[x1] += 1
            else:
                a = fractions.Fraction(y2 - y1, x2 - x1)
                b = y2 - x2 * a
                lines[a, b] += 1

        # if line contains n points
        # lines[a,b] = (n choose 2)
        # how to extract n?
        # x = n*(n-1) / 2
        # 2x = n*(n-1)
        # 2x = n^2 - n
        # 0 = n^2 - n - 2x
        # delta = 1 + 8x
        # n = (1 + sqrt(delta) / 2)

        import math

        def comb_to_n(combs):
            delta = 1 + 8*combs
            n = (1 + math.sqrt(delta)) / 2
            n = int(n)
            return n

        m = 1
        for v in vert_lines.values():
            v = comb_to_n(v)
            if v > m:
                m = v

        for (a, b), v in lines.items():
            v = comb_to_n(v)
            if v > m:
                m = v

        print(lines)
        print(vert_lines)

        return m
33335