<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:
    double minAreaFreeRect(vector<vector<int>>& points) {
        int n = points.size();
        set<pair<int, int>> st;
        double area = INT_MAX;
        for(auto it: points) {
            st.insert({it[0], it[1]});
        }
        for(int i = 0; i < n; i++) {
            for(int j = i + 1; j < n; j++) {
                for(int k = j + 1; k < n; k++) {
                    int x1 = points[i][0], y1 = points[i][1];
                    int x2 = points[j][0], y2 = points[j][1];
                    int x3 = points[k][0], y3 = points[k][1];
                    //check if three points are orthogonal, they are if product of their slopes is -1
                    //i.e (y2-y1)* (y3-y1) – (x2-x1) (x3-x1) = 0
                    if ((x2 - x1) * (x3 - x1) + (y2 - y1) * (y3 - y1) != 0) continue;
                    //find the 4th point through point of bisection
                    //(x1+x3)/ 2 = (x2+x4)/2
                    int x4 =  x2 + x3 - x1, y4 = y2 + y3 - y1;
                    //check if p4 is present and calculate area
                    if(st.find({x4,y4}) != st.end()) {
                        double len = pow(pow((x2-x1),2) + pow(y2-y1,2), 0.5);
                        double bth = pow(pow((x3-x1),2) + pow(y3-y1,2), 0.5);
                        area = min(area, len*bth);
                    }
                }
            }
        }
       return area == INT_MAX ? 0 : area;     
    }
};

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