<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>

#include <unordered_map>
#include <vector>
#include <cmath>
#include <algorithm>
using namespace std;

class Solution {
public:
    double minAreaFreeRect(vector<vector<int>>& points) {
        int n = points.size();
        unordered_map<string, vector<pair<int, int>>> midpointMap;
        double minArea = DBL_MAX;
        
        // Function to calculate the distance squared
        auto distSquared = [](const vector<int>& p1, const vector<int>& p2) {
            return (p1[0] - p2[0]) * (p1[0] - p2[0]) + (p1[1] - p2[1]) * (p1[1] - p2[1]);
        };
        
        // Create a map of diagonals with the same midpoint and distance squared
        for (int i = 0; i < n; ++i) {
            for (int j = i + 1; j < n; ++j) {
                // Calculate midpoint and distance squared
                int midX = points[i][0] + points[j][0];
                int midY = points[i][1] + points[j][1];
                int distSq = distSquared(points[i], points[j]);
                
                // Create a unique key for midpoint and distSq
                string key = to_string(midX) + "_" + to_string(midY) + "_" + to_string(distSq);
                midpointMap[key].emplace_back(i, j);
            }
        }
        
        // Iterate over all diagonals with the same midpoint and distance
        for (const auto& entry : midpointMap) {
            const vector<pair<int, int>>& diagonalPairs = entry.second;
            if (diagonalPairs.size() > 1) {
                for (int i = 0; i < diagonalPairs.size(); ++i) {
                    for (int j = i + 1; j < diagonalPairs.size(); ++j) {
                        // Get two diagonal pairs
                        auto [p1, p2] = diagonalPairs[i];
                        auto [p3, p4] = diagonalPairs[j];
                        
                        // Calculate the area using the cross product of vectors
                        // Area = 0.5 * |(x2 - x1) * (y4 - y3) - (y2 - y1) * (x4 - x3)|
                        double area = abs((points[p2][0] - points[p1][0]) * (points[p4][1] - points[p3][1]) - 
                                          (points[p2][1] - points[p1][1]) * (points[p4][0] - points[p3][0])) / 2.0;
                        
                        // Update the minimum area
                        if (area > 0) {
                            minArea = min(minArea, area);
                        }
                    }
                }
            }
        }
        
        return (minArea == DBL_MAX) ? 0 : minArea;
    }
};

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