A-Level Mathematics: Applications of Tangent and Chord Properties of Circles | A-Level 数学：圆的切线与弦的性质应用

📚 A-Level Mathematics: Applications of Tangent and Chord Properties of Circles | A-Level 数学：圆的切线与弦的性质应用

In this A-Level Mathematics revision article, we explore the tangent and chord properties of circles, a key topic in coordinate geometry and pure mathematics. These properties not only appear in dedicated circle questions but also form the foundation for solving problems involving intersections, distances, and equations of lines. Mastering them will help you handle both short-answer and extended-response questions with confidence.

本文为 A-Level 数学备考专题：圆的切线与弦的性质应用。该内容是坐标几何与纯数学的重要考点，不仅会在直接的圆类题目中出现，也是求解交点、距离与直线方程等问题的关键基础。掌握这些性质，你就能更加从容地应对选择题、简答题和证明题。

1. Core Concepts Review | 核心概念回顾

A circle is the set of all points in a plane that are equidistant from a fixed point called the centre. The fixed distance is the radius. A chord is a line segment joining two points on the circumference. A tangent is a line that touches the circle at exactly one point, called the point of tangency. The line segment from the centre to a point on the circumference is a radius, and a chord that passes through the centre is called a diameter.

圆是平面上到定点（圆心）的距离等于定长（半径）的所有点的集合。连接圆上任意两点的线段称为弦；与圆恰好只有一个公共点的直线称为切线，该公共点称为切点。圆心到圆上一点的线段是半径；经过圆心的弦称为直径，直径是圆内最长的弦。

• Radius: a line segment from the centre to a point on the circle.
• Chord: a line segment whose endpoints lie on the circle.
• Tangent: a line intersecting the circle at exactly one point.
• Point of tangency: the common point of a tangent and the circle.

• 半径：圆心到圆上一点的线段。
• 弦：两端点都在圆上的线段。
• 切线：与圆只有一个公共点的直线。
• 切点：切线与圆的唯一交点。

2. Tangent-Radius Theorem | 切线-半径定理

The tangent-radius theorem states that a tangent to a circle is perpendicular to the radius drawn to the point of tangency. This is one of the most important circle properties and is used repeatedly in coordinate geometry, trigonometry, and proof questions. If O is the centre, T is the point of tangency, and PT is a tangent, then OT ⊥ PT.

切线-半径定理：圆的切线垂直于经过切点的半径。这是圆的最重要性质之一，在坐标几何、三角学和证明题中被反复使用。若 O 为圆心，T 为切点，PT 为切线，则 OT ⊥ PT。

Example 1: A circle has centre O(0,0) and contains the point T(3,4). Find the slope of the tangent at T. The radius OT has slope 4/3. Since the tangent is perpendicular to OT, its slope is -3/4. Therefore the tangent equation is y – 4 = -3/4(x – 3), which simplifies to 3x + 4y = 25.

例 1：已知圆 O 的圆心为 O(0,0)，且过点 T(3,4)，求在 T 处的切线斜率。半径 OT 的斜率为 4/3。因为切线与 OT 垂直，所以切线斜率为 -3/4。于是切线方程为 y – 4 = -3/4(x – 3)，化简得 3x + 4y = 25。

This example demonstrates the standard method of moving from a radius to its tangent. In many exam problems, writing down the slope of the radius first and then taking the negative reciprocal is enough to unlock the whole solution.

这个例子展示了从半径到切线的标准转换方法。在许多考试题目中，先写出半径斜率，再取其负倒数，往往就能打开整个解题思路。

3. Equal Tangent Lengths | 切线长度相等

From a fixed external point P, two tangents can be drawn to a circle. The lengths of these two tangent segments are equal. In the diagram with centre O, tangents PA and PB touch the circle at A and B. Then PA = PB. This follows from the congruence of right triangles OAP and OBP, because OA = OB, OP is common, and both triangles have a right angle at the point of tangency.

从圆外一点 P 可作圆的两条切线。这两条切线段长度相等。设圆心为 O，PA 与 PB 分别切圆于 A 和 B，则 PA = PB。理由：直角三角形 OAP 与直角三角形 OBP 中，OA = OB，OP 公用，且 ∠OAP = ∠OBP = 90°，所以两直角三角形全等，故 PA = PB。

This property is often used to set up equations involving unknown lengths in geometry problems. For example, if a circle is inscribed in a triangle, the tangent lengths from each vertex to the points of contact can be expressed in terms of the side lengths, allowing the side lengths to be determined.

该性质常用于在几何问题中建立含未知长度的方程。例如，当圆内切于一个三角形时，从每个顶点到对应切点的切线长可以用边长表达，从而求出边长。

In coordinate geometry problems, once you have found one tangent length, the other tangent from the same external point has the same value, which can save time and simplify calculations.

在坐标几何问题中，一旦求出某条切线段的长度，从同一点引出的另一条切线段长度相同，这可以节省时间并简化计算。

4. Tangent-Chord Theorem | 切线-弦定理

The tangent-chord theorem, also called the alternate segment theorem, states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. If tangent AT touches the circle at A, and chord AB is drawn, then the angle between AT and AB equals the angle in the alternate segment, ∠ACB, where C is any point on the circumference on the opposite side of chord AB.

切线-弦定理（又称交替线段定理）：切线与过切点的弦所构成的角，等于该角的交替弧段内的圆周角。设切线 AT 切圆于 A，弦 AB 已作出，则 AT 与 AB 的夹角等于交替弧段内的圆周角 ∠ACB，其中 C 为 AB 对侧圆周上的任意一点。

Mathematically, ∠TAB = ∠ACB and ∠BAE = ∠ADB if AE is the other tangent direction. This theorem is essential for proving angle equalities in circle geometry and for solving problems that involve both tangents and chords.

数学上，有 ∠TAB = ∠ACB，若 AE 为切线另一方向，则 ∠BAE = ∠ADB。该定理在圆几何中用于证明角相等，以及解决涉及切线与弦的综合问题。

When using this theorem, always identify the two segments carefully: one ray of the

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