📚 Circle Theorems – A Complete Revision Guide | 圆定理 – 完整复习指南
Circle theorems are a set of geometric rules that describe the relationships between angles, chords, tangents, and arcs in a circle. They appear in virtually every IGCSE Mathematics extended paper, usually as a multi-part question worth 4 to 6 marks. Mastering these theorems unlocks a wide range of exam marks and provides a strong foundation for higher-level geometry.
圆定理是一组描述圆中角、弦、切线与弧之间关系的几何法则。它们几乎出现在每一份IGCSE数学扩展卷中，通常作为一道含多小问的题目，分值在4到6分之间。掌握这些定理能够帮助你在考试中获得大量分数，并为更高级的几何学习打下坚实基础。
1. Key Terms You Must Know | 必须掌握的关键术语
Before applying the theorems, you must be completely confident with the vocabulary below. Each theorem names a specific part of the circle, and misidentifying these parts is the most common source of errors.
在应用定理之前，你必须完全熟悉下列术语。每个定理都涉及圆的特定部分，而误判这些部分是学生最常犯的错误来源。
| Term | 术语 | Definition | 定义 |
| Centre (O) | 圆心(O) | The fixed inner point equidistant from every point on the circumference | 到圆周上每一点距离都相等的内点 |
| Radius | 半径 | A line from the centre to the circumference; plural is radii | 从圆心到圆周的线段；复数为radii |
| Chord | 弦 | A line segment joining two points on the circumference | 连接圆周上两点的线段 |
| Diameter | 直径 | A chord that passes through the centre; twice the radius | 经过圆心的弦；是半径的两倍 |
| Tangent | 切线 | A straight line that touches the circle at exactly one point | 与圆仅有一个接触点的直线 |
| Segment | 弓形 | The region between a chord and the arc it cuts off | 弦与其所截弧之间的区域 |
| Subtend | 所对 | To form an angle at a point from two given points | 从给定两点在某一点处形成一个角 |
Make a quick sketch of each term as you revise. Visual memory is essential for circle theorems because exam diagrams rarely label every part for you.
复习时请为每个术语快速画图。视觉记忆对圆定理至关重要，因为考试图形很少会为你标注所有部分。
2. Theorem 1: Angle at the Centre is Twice the Angle at the Circumference | 定理一：圆心角是圆周角的两倍
When two points A and B lie on a circle and O is the centre, the angle subtended at the centre by chord AB is exactly twice the angle subtended at the circumference by the same chord.
当圆周上有两点A和B、O为圆心时，弦AB在圆心处所对的角恰好是在圆周上同弦所对角的两倍。
∠AOB = 2 × ∠ACB
This theorem works for every point C on the circumference, provided C lies on the major arc when AOB is a minor arc, or on the minor arc when AOB is a major arc. In other words, C must be on the opposite arc to the one directly between A and B.
只要C位于与AB之间弧相对的那条弧上（即∠AOB为小弧时C在大弧上，∠AOB为大弧时C在小弧上），该定理对圆周上的任何点C都成立。
Worked example: O is the centre and ∠AOB = 140°. Find ∠ACB.
例题：O为圆心，∠AOB = 140°。求∠ACB。
Solution: By the angle at centre theorem, ∠ACB = 140° ÷ 2 = 70°.
解答：根据圆心角定理，∠ACB = 140° ÷ 2 = 70°。
Exam tip: If a diagram contains a double arc or a small circle marking at O, the examiner intends you to use this theorem first.
考试
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