<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>

import math
from collections import defaultdict

class Solution:
    def minAreaFreeRect(self, points: List[List[int]]) -> float:
        # Dictionary to store points based on their midpoints
    # Create a set of point tuples for O(1) look-up times

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

      

        # Get the number of points

        num_points = len(points)

      

        # Initialize answer as infinity to track minimum

        min_area = inf

      

        # Iterate over all points to check for rectangles

        for i in range(num_points):

            x1, y1 = points[i]

            for j in range(num_points):

                if j != i:

                    x2, y2 = points[j]

                    for k in range(j + 1, num_points):

                        if k != i:

                            x3, y3 = points[k]

                            # Calculate coordinates for the potential fourth point

                            x4 = x2 - x1 + x3

                            y4 = y2 - y1 + y3

                            # Check if the fourth point exists in the set

                            if (x4, y4) in point_set:

                                # Check for perpendicular vectors (dot product == 0)

                                vector_21 = (x2 - x1, y2 - y1)

                                vector_31 = (x3 - x1, y3 - y1)

                                if vector_21[0] * vector_31[0] + vector_21[1] * vector_31[1] == 0:

                                    # Calculate widths and heights of the rectangle

                                    width = sqrt(vector_21[0] ** 2 + vector_21[1] ** 2)

                                    height = sqrt(vector_31[0] ** 2 + vector_31[1] ** 2)

                                    # Update minimum area if a smaller one is found

                                    min_area = min(min_area, width * height)

        # Return 0 if no rectangle was found, else return the minimum area

        return 0 if min_area == inf else min_area
16500