⭕ 圆的方程：到圆心距离都相等⭕ Equations of Circles: All Points Equidistant from the Center
上一片里，两个点就定下了一条直线。这一片，小算盘机器人🧮 要画一个圆——它发现，圆其实是一句"距离都相等"翻译出来的。Last leaf, two points pinned a line. This leaf, robot 🧮 draws a circle — and finds it is just the sentence "everyone at the same distance" turned into math.
第 1 步：圆规转一圈，距离都一样Step 1: One turn of the compass
先看圆最朴素的样子：一根针，一支笔，转一圈。Start with the plainest circle: one pin, one pencil, one full turn.
圆规的针尖扎在圆心，笔尖到针尖的距离始终不变——转一圈，画出的每个点到圆心都一样远，这就是圆。换成坐标说：点到圆心的距离，就是勾股定理 √((x−a)²+(y−b)²)。The compass pin sits at the center and the pencil never changes its distance to it — turn once and every drawn point is equally far from the center. That is a circle. In coordinates that distance is the Pythagorean √((x−a)²+(y−b)²).
既然"距离都相等"就是圆，能不能把它直接写成方程？If "all distances equal" is what a circle is, can we write that as an equation?
第 2 步：距离公式平方一下Step 2: Square the distance
雷达站以自己为圆心扫出一圈圈绿波。设圆心是 (a, b)、半径是 r，圆上每个点都满足 √((x−a)²+(y−b)²) = r；两边平方，就是圆的方程 (x−a)²+(y−b)² = r²。圆心管位置、半径管大小——三个数字，圆就定了。The radar station sweeps green rings out from itself as the center. With center (a, b) and radius r, every point on a ring satisfies √((x−a)²+(y−b)²) = r; square both sides and you get the circle equation (x−a)²+(y−b)² = r². Center sets position, radius sets size — three numbers fix the circle.
这个式子不只管雷达，它管着所有圆——水里的也一样。This equation rules more than radar — look at a pond.
第 3 步：水波里的同一个方程Step 3: The same equation on water
石子落进池塘，水波从落点一圈圈扩开：每个波圈上，水到落点的距离都相等，所以每个波圈都是圆。要判断一个点在不在圆上，把它代进方程：距离算出来正好是 r，它就在圆上。A pebble drops into a pond and ripples spread from the splash: on each ring every drop of water is the same distance from the splash point, so every ring is a circle. To test whether a point lies on a circle, plug it into the equation: if its distance works out to exactly r, it is on.
🎮 你来当圆心（30 秒）🎮 Your turn: be the center (30 seconds)
下面每关有两个金色亮点：按住画布拖动圆心，再调好半径，让圆同时穿过两个点。Each level has two golden points: drag the center on the canvas, tune the radius, and make the circle pass through both.
一句话记住它：圆的方程就是距离公式的平方版——(x−a)²+(y−b)²=r²；到圆心 (a, b) 的距离恰好为 r 的点，全在圆上。Remember it in one line: the circle equation is the distance formula squared — (x−a)²+(y−b)²=r², collecting every point whose distance to the center (a, b) is exactly r.
圆 = 到圆心距离都相等的点集：圆心管位置，半径管大小A circle is the set of points equally far from its center: center sets position, radius sets size(x−a)²+(y−b)²=r² 只是把距离公式两边平方，去掉了根号(x−a)²+(y−b)²=r² is just the distance formula squared — no more square root判断点在不在圆上：代入一算，距离等于 r 就在To test a point: plug it in — distance equal to r means it's on the circle
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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