<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>

const static auto speedup = [] () {
    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr);
    std::cout.tie(nullptr);
    
    return 1;
} ();

class Solution {
public:
size_t d2(int x1, int y1, int x2, int y2) { 
    return (x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2);
}
double minAreaFreeRect(vector<vector<int>>& ps, size_t res = 0) {
  unordered_map<size_t, vector<vector<int>>> m;
  for (auto i = 0; i < ps.size(); ++i)
    for (auto j = i + 1; j < ps.size(); ++j) {
      auto center = ((size_t)(ps[i][0] + ps[j][0]) << 16) + ps[i][1] + ps[j][1];
      m[center].push_back({ ps[i][0], ps[i][1], ps[j][0], ps[j][1] });
    }
  for (auto it = begin(m); it != end(m); ++it)
    for (auto i = 0; i < it->second.size(); ++i)
      for (auto j = i + 1; j < it->second.size(); ++j) {
        auto &p1 = it->second[i], &p2 = it->second[j];
        if ((p1[0] - p2[0]) * (p1[0] - p2[2]) + (p1[1] - p2[1]) * (p1[1] - p2[3]) == 0) {
          auto area = d2(p1[0], p1[1], p2[0], p2[1]) * d2(p1[0], p1[1], p2[2], p2[3]);
          if (res == 0 || res > area) res = area;
        }
      }
  return sqrt(res);
}

private:
    int dist(
        vector<int>& p1,
        vector<int>& p2
    ) {
        return (p1[0] - p2[0]) * (p1[0] - p2[0]) + 
            (p1[1] - p2[1]) * (p1[1] - p2[1]);
    }

    pair<double, bool> helper(
        vector<int>& p1,
        vector<int>& p2,
        vector<int>& p3,
        vector<int>& p4
    ) {
        int d12 = dist(p1, p2), 
            d13 = dist(p1, p3), 
            d14 = dist(p1, p4);

        int diag = std::max({d12, d13, d14});
        vector<int> dgpt, pt1, pt2;
        if (diag == d12) {
            dgpt = p2, pt1 = p3, pt2 = p4;
        } else if (diag == d13) {
            dgpt = p3, pt1 = p2, pt2 = p4;
        } else {
            dgpt = p4, pt1 = p2, pt2 = p3;
        }

        auto diag2 = dist(pt1, pt2);

        if (diag2 != diag) return {-1, false};
        if (dist(p1, pt1) != dist(dgpt, pt2)) return {-1, false};
        if (dist(p1, pt2) != dist(dgpt, pt1)) return {-1, false};

        return {sqrt((double) dist(p1, pt1)) * sqrt((double) dist(p1, pt2)), true};
    }
};
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