Maximum Number of Darts Inside of a Circular Dartboard
<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://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/1400-1499/1453.Maximum%20Number%20of%20Darts%20Inside%20of%20a%20Circular%20Dartboard/images/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://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/1400-1499/1453.Maximum%20Number%20of%20Darts%20Inside%20of%20a%20Circular%20Dartboard/images/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>
Geometry; Array; Math
class Solution:
    def numPoints(self, darts: list[list[int]], r: int) -> int:
        def countDarts(x, y):
            count = 0
            for x1, y1 in darts:
                if dist((x, y), (x1, y1)) <= r + 1e-7:
                    count += 1
            return count

        def possibleCenters(x1, y1, x2, y2):
            dx, dy = x2 - x1, y2 - y1
            d = sqrt(dx * dx + dy * dy)
            if d > 2 * r:
                return []
            mid_x, mid_y = (x1 + x2) / 2, (y1 + y2) / 2
            dist_to_center = sqrt(r * r - (d / 2) * (d / 2))
            offset_x = dist_to_center * dy / d
            offset_y = dist_to_center * -dx / d
            return [
                (mid_x + offset_x, mid_y + offset_y),
                (mid_x - offset_x, mid_y - offset_y),
            ]

        n = len(darts)
        max_darts = 1

        for i in range(n):
            for j in range(i + 1, n):
                centers = possibleCenters(
                    darts[i][0], darts[i][1], darts[j][0], darts[j][1]
                )
                for center in centers:
                    max_darts = max(max_darts, countDarts(center[0], center[1]))

        return max_darts
