<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:
    min_area = float("inf")
    seen = dict()
    for i, (x1,y1) in enumerate(points):
      for x2,y2 in points[i+1:]:
        cx = (x1+x2)/2
        cy = (y1+y2)/2
        d = (x1-x2)**2 + (y1-y2)**2

        for xi,yi in seen.setdefault((cx,cy,d),[]):
          side1 = (x1-xi)**2 + (y1-yi)**2
          side2 = (x2-xi)**2 + (y2-yi)**2
          min_area = min(min_area, (side1*side2)**0.5)
        seen.setdefault((cx,cy,d),[]).append((x1,y1))
    return 0 if min_area == float("inf") else min_area

  def minAreaFreeRect_v1(self, points: List[List[int]]) -> float:
    """ O(N^4) """
    min_area = float("inf")
    for x1,y1 in points:
      for x2,y2 in points:
        if [x2,y2] == [x1,y1]:
          continue
        for x3,y3 in points:
          if [x3,y3] in [[x1,y1],[x2,y2]]:
            continue
          P = (x1-x2)**2 + (y1-y2)**2
          B = (x1-x3)**2 + (y1-y3)**2
          H = (x2-x3)**2 + (y2-y3)**2
          if P + B == H:
            for x4,y4 in points:
              if [x4,y4] not in [[x1,y1],[x2,y2],[x3,y3]]:
                P2 = (x4-x2)**2 + (y4-y2)**2
                B2 = (x4-x3)**2 + (y4-y3)**2
                H2 = (x2-x3)**2 + (y2-y3)**2
                H3 = (x1-x4)**2 + (y1-y4)**2

                chk = (P == P2 and B==B2) or (P==B2 and B==P2) 
                chk = chk and H==H2==H3
                if chk and P2+B2 == H2:
                  min_area = min(min_area, (P*B)**0.5)

    return 0 if min_area == float("inf") else min_area
17300