📚 Use Tangent and Chord Properties | 使用切线与弦性质
For Edexcel A-Level Mathematics, questions involving circles often require you to apply tangent and chord properties in both pure geometry and coordinate geometry. These properties connect the centre, the radius, the tangent, the chord and the angle relationships inside a circle, and they appear regularly in proof questions, in equation-of-a-tangent questions and in longer coordinate geometry problems where a circle is only one part of the story. This guide collects every property you are expected to know, shows how each one is used in exam-style working, and finishes with a checklist you can run through before the exam.
在 Edexcel A-Level 数学中，涉及圆的题目经常要求你在纯几何与坐标几何中应用切线与弦的性质。这些性质把圆心、半径、切线、弦以及圆内的角度关系联系在一起，经常出现在证明题、求切线方程的题目，以及圆只是其中一环的较长坐标几何综合题中。本文系统整理你需要掌握的全部相关性质，演示它们在各考试题型中的具体用法，并在结尾给出考前检查清单。
1. Circle Vocabulary and Key Facts | 圆的基本术语与关键事实
Every circle question is written in a small vocabulary. Being fluent in it prevents the most common misreading, which is confusing a tangent with a chord or a radius with a diameter. Before applying any theorem, make sure you are confident with the standard language used in circle questions.
每一道圆的题目都是用一套不大的术语体系写成的。熟悉这套术语，可以避免最常见的误读——把切线与弦混为一谈，或者把半径当成直径。在应用任何定理之前，请确保你熟悉圆类题目中常用的标准术语。
- Centre O: the point equidistant from every point on the circle — 圆心 O：到圆上每一点距离都相等的点
- Radius r: a segment from the centre to any point on the circle — 半径 r：从圆心到圆上任意一点的线段
- Chord: a segment joining two points on the circle — 弦：连接圆上两点的线段
- Diameter: a chord that passes through the centre — the longest possible chord — 直径：经过圆心的弦，也是圆中最长的弦
- Tangent: a line that touches the circle at exactly one point — 切线：与圆恰好只有一个公共点的直线
- Secant: a line that cuts the circle at two points — 割线：与圆相交于两点的直线
- Point of contact: the single point where a tangent meets the circle — 切点：切线与圆唯一接触的点
- External point: a point outside the circle, usually called T or P in questions — 外点：位于圆外的点，题目中通常记为 T 或 P
One structural fact matters more than it looks: a tangent is a line, while a chord is a segment. A chord always lies inside or on the circle, and a tangent never passes through the interior. Many marks are lost simply because a student treats a tangent as if it were a chord and applies the wrong theorem to it.
有一个结构性区别比看上去更重要：切线是一条直线，而弦是一条线段。弦始终位于圆内部或圆上，切线则永远不会穿过圆的内部。很多分数就是因为把切线当成弦、套用了错误的定理而丢掉的。
2. Tangent is Perpendicular to the Radius | 切线垂直于半径
The central property for tangents is: at the point of contact, the tangent is perpendicular to the radius drawn to that point. If the centre is O and P is the point of contact, then the radius OP and the tangent line meet at 90°.
切线最核心的性质是：在切点处，切线与经过该点的半径垂直。如果圆心为 O，切点为 P，那么半径 OP 与切线相交成 90°。
OP ⊥ tangent at P
This single fact generates almost every tangent calculation you will meet. It converts a geometric statement into an algebraic one, because perpendicular lines have gradients whose product is −1.
这一条事实几乎衍生出你遇到的所有切线计算。它把一个几何陈述转化为代数陈述，因为互相垂直的两条直线，其斜率之积为 −1。
mr × mt = −1
Here mr is the gradient of the radius and mt izontal, the tangent is vertical and its gradient is undefined, so write the tangent as x = a rather than trying to use the product rule.
其中 mr 是半径的斜率，mt 是切线的斜率。要注意特殊情况：若半径竖直，则切线水平，斜率为 0；若半径水平，则切线竖直、斜率不存在，此时应把切线写成 x = a，而不是硬套乘积等于 −1 的公式。
The converse is also true and is useful in proofs: if a line through a point on the circle is perpendicular to the radius at that point, then the line is a tangent to the circle. This is how you prove tangency when you are given an equation of a line rather than a picture.
逆命题同样成立，在证明题中很有用：若过圆上某点的直线与该点处的半径垂直，则该直线是圆的切线。当你拿到的是直线方程而不是图形时，就是靠这一条来证明相切的。
3. Equation of a Tangent in Coordinate Geometry | 坐标几何中的切线方程
b)² = r². To find the tangent at a point P(x₁, y₁) on the circle, follow a fixed routine.
对于圆心为 (a, b)、半径为 r 的圆，其方程为 (x − a)² + (y − b)² = r²。要求圆上一点 P(x₁, y₁) 处的切线，按固定步骤操作即可。
- Step 1: read off the centre (a, b) and the radius r — 第一步：读出圆心 (a, b) 与半径 r
- Step 2: verify that P really lies on the circle by substituting its coordinates — 第二步：把 P 的坐标代入方程，验证它确实在圆上
- Step 3: find the gradient of the radius OP — 第三步：求半径 OP 的斜率
- Step 4: take the negative reciprocal to get the tangent gradient — 第四步：取负倒数，得到切线斜率
- Step 5: use y − y₁ = m(x − x₁) and rearrange into the form ax + by + c = 0 — 第五步：用点斜式 y − y₁ = m(x − x₁)，再整理成 ax + by + c = 0 的形式
Worked example. (y + 2)² = 25 and the point P(6, 2) lies on C. Find the equation of the tangent to C at P.
例题。 圆 C 的方程为 (x − 3)² + (y + 2)² = 25，点 P(6, 2) 在 C 上。求 C 在 P 处的切线方程。
. Check P: (6 − 3)² + (2 + 2)² = 9 + 16 = 25, so P is on the circle. Gradient of radius: mr = (2 − (−2)) ÷ (6 − 3) = 4 ÷ 3. Tangent gradient: mt = −3 ÷ 4. Tangent: y − 2 = (−3 ÷ 4)(x − 6), which rearranges to 3x + 4y = 26.
圆心为 (3, −2)，半径为 5。检验 P：(6 − 3)² + (2 + 2)² = 9 + 16 = 25，故 P 在圆上。半径斜率 mr = (2 − (−2)) ÷ (6 − 3) = 4 ÷ 3。切线斜率 mt = −3 ÷ 4。切线方程：y − 2 = (−3 ÷ 4)(x − 6)，整理得 3x + 4y = 26。
Always self-check with the distance formula: the perpendicular distance from the centre to the tangent must equal r. Here |3(3) + 4(−2) − 26| ÷ √9 + 16 = 25 ÷ 5 = 5, which matches the radius.
一定要用点到直线距离公式自检：圆心到切线的距离必须等于半径。这里 |3(3) + 4(−2) − 26| ÷ √9 + 16 = 25 ÷ 5 = 5，与半径相符。
4. Two Tangents From an External Point | 从外点引出的两条切线
From a point T outside a circle you can draw exactly two tangents. If they touch the circle at A and B, then the two tangent lengths are equal, and the line OT bisects both the angle between the tangents and the angle between the radii.
从圆外一点 T 可以恰好作两条切线。若它们分别切圆于 A、B，则两条切线长度相等，且直线 OT 同时平分两切线之间的夹角与两半径之间的夹角。
TA = TB and ∠ATO = ∠BTO
worth knowing: triangles OAT and OBT are congruent because OA = OB = r, OT is common, and ∠OAT = ∠
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