<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 {
public:

struct Hasher {
    size_t operator()(const vector<int>& a) const {
        uint64_t h = a[0];
        h = h << 32 | a[1];
        return hash<uint64_t>{}(h);
    }
};

double minAreaFreeRect(vector<vector<int>>& points) {
    const int N = points.size();

    // O(N log N)
    sort(points.begin(), points.end(), [](const vector<int>& l, const vector<int>& r) {
        if (l[0] == r[0]) return l[1] < r[1];
        else return l[0] < r[0];
    });

    unordered_set<vector<int>, Hasher> hpoints;
    for (int i=0; i<N; i++) {
        hpoints.emplace(points[i]);
    }

    double min_area = double(INT_MAX);

    // O(NNN)
    for (int i=0; i<N; i++) {
        for (int j=i+1; j<N; j++) {
            for (int k=j+1; k<N; k++) {
                int ax = points[i][0];
                int ay = points[i][1];

                int bx = points[j][0];
                int by = points[j][1];

                int cx = points[k][0];
                int cy = points[k][1];

                int ab_dx = bx - ax;
                int ab_dy = by - ay;

                int ac_dx = cx - ax;
                int ac_dy = cy - ay;

                // ab x bc = 0
                if ((ab_dx * ac_dx + ab_dy * ac_dy) != 0) continue;

                // delta_ab = delta_cd
                // dab = d - c -> d = dba + c

                int dx = ab_dx + cx;
                int dy = ab_dy + cy;

                if (hpoints.find({dx, dy}) == hpoints.end()) continue;

                //double area = (double)abs(ab_dx * ac_dy - ab_dy * ac_dx);
                double len_ab = (ab_dx*ab_dx + ab_dy*ab_dy);
                double len_ac = (ac_dx*ac_dx + ac_dy*ac_dy);
                double area = sqrt(len_ab) * sqrt(len_ac);
                min_area = min(min_area, area);
            }
        }
    }

    if (min_area == double(INT_MAX)) return 0.0;

    return min_area;
}

};
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