Check if the Rectangle Corner Is Reachable
<p>You are given two positive integers <code>xCorner</code> and <code>yCorner</code>, and a 2D array <code>circles</code>, where <code>circles[i] = [x<sub>i</sub>, y<sub>i</sub>, r<sub>i</sub>]</code> denotes a circle with center at <code>(x<sub>i</sub>, y<sub>i</sub>)</code> and radius <code>r<sub>i</sub></code>.</p>

<p>There is a rectangle in the coordinate plane with its bottom left corner at the origin and top right corner at the coordinate <code>(xCorner, yCorner)</code>. You need to check whether there is a path from the bottom left corner to the top right corner such that the <strong>entire path</strong> lies inside the rectangle, <strong>does not</strong> touch or lie inside <strong>any</strong> circle, and touches the rectangle <strong>only</strong> at the two corners.</p>

<p>Return <code>true</code> if such a path exists, and <code>false</code> otherwise.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">xCorner = 3, yCorner = 4, circles = [[2,1,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">true</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/3200-3299/3235.Check%20if%20the%20Rectangle%20Corner%20Is%20Reachable/images/example2circle1.png" style="width: 346px; height: 264px;" /></p>

<p>The black curve shows a possible path between <code>(0, 0)</code> and <code>(3, 4)</code>.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">xCorner = 3, yCorner = 3, circles = [[1,1,2]]</span></p>

<p><strong>Output:</strong> <span class="example-io">false</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/3200-3299/3235.Check%20if%20the%20Rectangle%20Corner%20Is%20Reachable/images/example1circle.png" style="width: 346px; height: 264px;" /></p>

<p>No path exists from <code>(0, 0)</code> to <code>(3, 3)</code>.</p>
</div>

<p><strong class="example">Example 3:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">xCorner = 3, yCorner = 3, circles = [[2,1,1],[1,2,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">false</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/3200-3299/3235.Check%20if%20the%20Rectangle%20Corner%20Is%20Reachable/images/example0circle.png" style="width: 346px; height: 264px;" /></p>

<p>No path exists from <code>(0, 0)</code> to <code>(3, 3)</code>.</p>
</div>

<p><strong class="example">Example 4:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">xCorner = 4, yCorner = 4, circles = [[5,5,1]]</span></p>

<p><strong>Output:</strong> <span class="example-io">true</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/3200-3299/3235.Check%20if%20the%20Rectangle%20Corner%20Is%20Reachable/images/rectangles.png" style="width: 346px; height: 264px;" /></p>
</div>

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

<ul>
	<li><code>3 &lt;= xCorner, yCorner &lt;= 10<sup>9</sup></code></li>
	<li><code>1 &lt;= circles.length &lt;= 1000</code></li>
	<li><code>circles[i].length == 3</code></li>
	<li><code>1 &lt;= x<sub>i</sub>, y<sub>i</sub>, r<sub>i</sub> &lt;= 10<sup>9</sup></code></li>
</ul>
Depth-First Search; Breadth-First Search; Union Find; Geometry; Array; Math
class Solution:
    def canReachCorner(
        self, xCorner: int, yCorner: int, circles: List[List[int]]
    ) -> bool:
        def in_circle(x: int, y: int, cx: int, cy: int, r: int) -> int:
            return (x - cx) ** 2 + (y - cy) ** 2 <= r**2

        def cross_left_top(cx: int, cy: int, r: int) -> bool:
            a = abs(cx) <= r and 0 <= cy <= yCorner
            b = abs(cy - yCorner) <= r and 0 <= cx <= xCorner
            return a or b

        def cross_right_bottom(cx: int, cy: int, r: int) -> bool:
            a = abs(cx - xCorner) <= r and 0 <= cy <= yCorner
            b = abs(cy) <= r and 0 <= cx <= xCorner
            return a or b

        def dfs(i: int) -> bool:
            x1, y1, r1 = circles[i]
            if cross_right_bottom(x1, y1, r1):
                return True
            vis[i] = True
            for j, (x2, y2, r2) in enumerate(circles):
                if vis[j] or not ((x1 - x2) ** 2 + (y1 - y2) ** 2 <= (r1 + r2) ** 2):
                    continue
                if (
                    (x1 * r2 + x2 * r1 < (r1 + r2) * xCorner)
                    and (y1 * r2 + y2 * r1 < (r1 + r2) * yCorner)
                    and dfs(j)
                ):
                    return True
            return False

        vis = [False] * len(circles)
        for i, (x, y, r) in enumerate(circles):
            if in_circle(0, 0, x, y, r) or in_circle(xCorner, yCorner, x, y, r):
                return False
            if (not vis[i]) and cross_left_top(x, y, r) and dfs(i):
                return False
        return True
