<p>Alice is throwing <code>n</code> darts on a very large wall. You are given an array <code>darts</code> where <code>darts[i] = [x<sub>i</sub>, y<sub>i</sub>]</code> is the position of the <code>i<sup>th</sup></code> dart that Alice threw on the wall.</p>

<p>Bob knows the positions of the <code>n</code> darts on the wall. He wants to place a dartboard of radius <code>r</code> on the wall so that the maximum number of darts that Alice throws lie&nbsp;on the dartboard.</p>

<p>Given the integer <code>r</code>, return <em>the maximum number of darts that can lie on the dartboard</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/04/29/sample_1_1806.png" style="width: 248px; height: 211px;" />
<pre>
<strong>Input:</strong> darts = [[-2,0],[2,0],[0,2],[0,-2]], r = 2
<strong>Output:</strong> 4
<strong>Explanation:</strong> Circle dartboard with center in (0,0) and radius = 2 contain all points.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/04/29/sample_2_1806.png" style="width: 306px; height: 244px;" />
<pre>
<strong>Input:</strong> darts = [[-3,0],[3,0],[2,6],[5,4],[0,9],[7,8]], r = 5
<strong>Output:</strong> 5
<strong>Explanation:</strong> Circle dartboard with center in (0,4) and radius = 5 contain all points except the point (7,8).
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= darts.length &lt;= 100</code></li>
	<li><code>darts[i].length == 2</code></li>
	<li><code>-10<sup>4</sup> &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 10<sup>4</sup></code></li>
	<li>All the <code>darts</code>&nbsp;are unique</li>
	<li><code>1 &lt;= r &lt;= 5000</code></li>
</ul>

class Solution {
public:
    int numPoints(vector<vector<int>>& darts, int r) {
        auto distance = [](double x1, double y1, double x2, double y2) {
            return sqrt(pow(x1 - x2, 2) + pow(y1 - y2, 2));
        };

        auto getCircleCenters = [&distance](double x1, double y1, double x2, double y2, int r) {
            vector<pair<double, double>> centers;
            double mx = (x1 + x2) / 2, my = (y1 + y2) / 2;
            double d = distance(x1, y1, x2, y2);
            
            if (d > 2 * r) return centers;
            
            double h = sqrt(r * r - pow(d/2, 2));
            double dx = (y2 - y1) / d, dy = (x1 - x2) / d;
            
            centers.push_back({mx + h*dx, my + h*dy});
            centers.push_back({mx - h*dx, my - h*dy});
            return centers;
        };

        int n = darts.size();
        int maxDarts = 1;
        
        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                double x1 = darts[i][0], y1 = darts[i][1];
                double x2 = darts[j][0], y2 = darts[j][1];
                
                auto centers = getCircleCenters(x1, y1, x2, y2, r);
                
                for (const auto& center : centers) {
                    double cx = center.first, cy = center.second;
                    int count = 0;
                    for (const auto& dart : darts) {
                        if (distance(dart[0], dart[1], cx, cy) <= r + 1e-8) {
                            count++;
                        }
                    }
                    maxDarts = max(maxDarts, count);
                }
            }
        }
        
        return maxDarts;
    }
};
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