<p>You are given an array of points in the <strong>X-Y</strong> plane <code>points</code> where <code>points[i] = [x<sub>i</sub>, y<sub>i</sub>]</code>.</p>

<p>Return <em>the minimum area of any rectangle formed from these points, with sides <strong>not necessarily parallel</strong> to the X and Y axes</em>. If there is not any such rectangle, return <code>0</code>.</p>

<p>Answers within <code>10<sup>-5</sup></code> of the actual answer will be accepted.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/21/1a.png" style="width: 398px; height: 400px;" />
<pre>
<strong>Input:</strong> points = [[1,2],[2,1],[1,0],[0,1]]
<strong>Output:</strong> 2.00000
<strong>Explanation:</strong> The minimum area rectangle occurs at [1,2],[2,1],[1,0],[0,1], with an area of 2.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/22/2.png" style="width: 400px; height: 251px;" />
<pre>
<strong>Input:</strong> points = [[0,1],[2,1],[1,1],[1,0],[2,0]]
<strong>Output:</strong> 1.00000
<strong>Explanation:</strong> The minimum area rectangle occurs at [1,0],[1,1],[2,1],[2,0], with an area of 1.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/22/3.png" style="width: 383px; height: 400px;" />
<pre>
<strong>Input:</strong> points = [[0,3],[1,2],[3,1],[1,3],[2,1]]
<strong>Output:</strong> 0
<strong>Explanation:</strong> There is no possible rectangle to form from these points.
</pre>

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

<ul>
	<li><code>1 &lt;= points.length &lt;= 50</code></li>
	<li><code>points[i].length == 2</code></li>
	<li><code>0 &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 4 * 10<sup>4</sup></code></li>
	<li>All the given points are <strong>unique</strong>.</li>
</ul>


class Solution:
    def minAreaFreeRect(self, points: List[List[int]]) -> float:
        def distance(p1, p2):
            return (p1[0] - p2[0]) ** 2 + (p1[1] - p2[1]) ** 2
        
        mid_diagonals = {}
        
        for i in range(len(points)):
            for j in range(i + 1, len(points)):
                mid_x = (points[i][0] + points[j][0]) / 2
                mid_y = (points[i][1] + points[j][1]) / 2
                dist = distance(points[i], points[j])
                
                if (mid_x, mid_y, dist) not in mid_diagonals:
                    mid_diagonals[(mid_x, mid_y, dist)] = []
                
                mid_diagonals[(mid_x, mid_y, dist)].append((points[i], points[j]))
        
        min_area = inf
        
        for diag_pairs in mid_diagonals.values():
            if len(diag_pairs) > 1:
                for (p1, p2) in diag_pairs:
                    for (p3, p4) in diag_pairs:
                        if p1 != p3 and p1 != p4 and p2 != p3 and p2 != p4:
                            area = sqrt(distance(p1, p3)) * sqrt(distance(p1, p4))
                            min_area = min(min_area, area)
        
        return 0 if min_area == inf else min_area

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