<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 <vector>
#include <map>
#include <tuple>
#include <cmath>
#include <limits>
using namespace std;

class Solution {
public:
    double dist_squared(int x1, int y1, int x2, int y2) {
        return (x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2);
    }

    double minAreaFreeRect(vector<vector<int>>& points) {
        map<string, vector<pair<int, int>>> pointsSet;
        int n = points.size();
        double minArea = numeric_limits<double>::infinity();

        for (int i = 0; i < n; i++) {
            int x0 = points[i][0], y0 = points[i][1];
            for (int j = i + 1; j < n; j++) {
                int x1 = points[j][0], y1 = points[j][1];
                int midX = x0 + x1, midY = y0 + y1;  // Use (2*midX, 2*midY) to avoid floats
                int d2 = dist_squared(x0, y0, x1, y1);

                string key = to_string(midX) + to_string(midY) + to_string(d2);
                if (pointsSet.find(key) != pointsSet.end()) {
                    for (const pair<int, int>& p : pointsSet[key]) {
                        int x2 = p.first, y2 = p.second;
                        double area = sqrt(dist_squared(x0, y0, x2, y2)) * sqrt(dist_squared(x1, y1, x2, y2));
                        minArea = min(minArea, area);
                    }
                }
                pointsSet[key].emplace_back(x0, y0);
            }
        }

        return minArea == numeric_limits<double>::infinity() ? 0 : minArea;
    }
};

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