<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 dist_pow2(x1, y1, x2, y2): return (x2 - x1)**2 + (y2 - y1)**2
        def dist(x1, y1, x2, y2): return dist_pow2(x1, y1, x2, y2)**0.5
        
        L, m = len(points), float('inf')

        quick_search = {(x, y) for x, y in points}

        for i in range(L):
            for j in range(i + 1, L):
                for k in range(j + 1, L):
                    x1, y1 = points[i] # calculate point4 to form a diamond by previous three
                    x2, y2 = points[j] # we only check case when x4 is latter occured than x3
                    x3, y3 = points[k] # becourse the case when x4 is earlier occured than x3 will be covered already
                    x4, y4 = (x3 + (x2 - x1), y3 + (y2 - y1))

                    if (x4, y4) in quick_search:  # look for existence of dimond shape
                        # if two diagonals' length are equal, we got a rectangel
                        if dist_pow2(x1, y1, x4, y4) == dist_pow2(x2, y2, x3, y3):
                            # calculate area by multipling any two adjacent edge's length
                            area = dist(x1, y1, x2, y2) * dist(x1, y1, x3, y3)
                            m = min(m, area)  # keep track of min_area

        return m if m != float('inf') else 0
                    
16700