IGCSE Maths Circle Theorems: Complete Revision Guide | IGCSE 数学圆定理：完整复习指南

📚 IGCSE Maths Circle Theorems: Complete Revision Guide | IGCSE 数学圆定理：完整复习指南

Circle theorems are a key part of the IGCSE Mathematics syllabus. They describe angle and length relationships within circles, and they appear frequently in both Paper 2 and Paper 4. Building a clear understanding of these rules will help you solve geometry problems confidently.

圆定理是 IGCSE 数学大纲中的关键内容。它们描述了圆内角与线段之间的关系，在 Paper 2 和 Paper 4 中频繁出现。清楚理解这些规则将帮助你自信地解决几何问题。

This guide covers the main circle theorems required for IGCSE, including angle at the centre, angles in the same segment, cyclic quadrilaterals, tangent properties, and the alternate segment theorem. Each section gives the rule, the common notation, and the typical exam clues you should look for.

本指南涵盖了 IGCSE 要求的主要圆定理，包括圆心角定理、同弦圆周角、圆内接四边形、切线性质以及弦切角定理。每一节都会给出规则、常用符号以及你应该寻找的典型考试线索。

1. Key Terms and Notation | 关键术语与符号

Before using circle theorems, make sure you know the basic terms: chord, tangent, arc, segment, sector, circumference, radius, and diameter. A chord is a line segment joining two points on the circle, while a tangent touches the circle at exactly one point.

在使用圆定理之前，请确保你熟悉基本术语：弦、切线、弧、弓形、扇形、圆周、半径和直径。弦是圆上两点之间的线段，而切线与圆只有一个接触点。

In diagrams, angles are usually labelled with three letters, and points on the circumference are often denoted by capital letters. You should also be comfortable using the symbols ° for degrees and ∠ for angle. For example, ∠APB means the angle at point P, formed by lines PA and PB.

在图中，角通常用三个字母标记，圆周上的点常用大写字母表示。你还应熟练使用度数符号 ° 和角符号 ∠。例如，∠APB 表示在点 P 处由线段 PA 和 PB 形成的角。

When a question says 'points A, B, C and D lie on the circumference', this usually signals that circle theorems will be needed. Always mark the centre O if it is given, because it is essential for the angle at the centre theorem.

当题目说“点 A、B、C、D 在圆周上”时，这通常提示你需要使用圆定理。如果给出了圆心 O，一定要标记出来，因为圆心角定理离不开它。

2. Angle at the Centre Theorem | 圆心角定理

The angle at the centre of a circle is twice any angle at the circumference subtended by the same arc.

圆心角等于同弧所对圆周角的两倍。

∠AOB = 2 × ∠APB

Here, O is the centre, and P is any point on the circumference standing on the same arc AB. The word 'subtended' means that the arms of the angle reach out to the two endpoints of the arc.

这里 O 是圆心，P 是圆周上与弧 AB 对应的任意点。“所对”是指角的两条边延伸到弧的两个端点。

For example, if the angle at the centre ∠AOB is 120°, then any angle at the circumference subtended by the same arc AB will be 60°. This theorem works for both the major arc and the minor arc, so be careful which arc the question is referring to.

例如，如果圆心角 ∠AOB 为 120°，那么同一条弧 AB 所对的任意圆周角都是 60°。这个定理对优弧和劣弧都成立，所以要仔细分清题目指的是哪一段弧。

3. Angles in the Same Segment | 同弦上的圆周角

Angles subtended by the same chord or arc in the same segment of a circle are equal.

在同一条弦或同一条弧的同一弓形中，所对的圆周角相等。

∠APB = ∠AQB

This is very useful when you see two angles opening onto the same chord from the same side of the chord. Points P and Q must lie in the same segment, meaning they are on the same side of chord AB.

当你看到两个角从弦的同一侧张向同一条弦时，这个定理非常有用。点 P 和 Q 必须位于同一弓形中，也就是说它们必须在弦 AB 的同一侧。

A common exam clue is a diagram with two triangles sharing the same base chord. Once you spot that the angles are in the same segment, you can write them as equal straight away. Do not confuse this with the angle at the centre theorem: here the two angles are equal to each other, not double.

常见的考试线索是图中两个三角形共用同一条弦作为底边。一旦你发现这些角在同一弓形中，就可以直接写出它们相等。不要把这个定理与圆心角定理混淆：这里两个角彼此相等，而不是两倍关系。

4. Angle in a Semicircle | 半圆上的圆周角

An angle subtended by a diameter at the circumference is always a right angle.

直径所对的圆周角总是直角。

If AB is a diameter, then ∠APB = 90°

This is a special case of the angle at the centre theorem: the diameter AB is a straight line through the centre, so the central angle ∠AOB is 180°. Therefore the angle at the circumference is half of 180°, which is 90°.

这是圆心角定理的特例：直径 AB 是经过圆心的直线，所以圆心角 ∠AOB 为 180°。因此圆周角是其一半，即 90°。

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Whenever you see a diameter in a circle, draw a triangle with the diameter as one side and mark the opposite angle as a right angle. This often allows you to use Pythagoras' theorem or trigonometric ratios in the resulting right-angled triangle.

每当你在圆中看到直径时，以直径为一边画出三角形，并把对角标为直角。这通常能让你在得到的直角三角形中使用勾股定理或三角比。

5. Cyclic Quadrilaterals | 圆内接四边形

A cyclic quadrilateral is a four-sided shape with all four vertices lying on the circumference of a circle. Its opposite angles add up to 180°.

圆内接四边形是指四个顶点都在同一个圆周上的四边形。它的对角之和为 180°。

∠A + ∠C = 180° and ∠B + ∠D = 180°

You can also use the exterior angle property: an exterior angle of a cyclic quadrilateral equals the interior opposite angle.

你还可以使用外角性质：圆内接四边形的一个外角等于其内对角。

For example, if side AD is extended to point E, then the exterior angle ∠CDE equals the interior opposite angle ∠B. This property is particularly handy when a question asks you to prove that two angles are equal.

例如，如果边 AD 延长到点 E，那么外角 ∠CDE 等于内对角 ∠B。当题目要求你证明两个角相等时，这个性质特别有用。

Remember that not every quadrilateral inside a circle is cyclic in the formal sense unless all four vertices are confirmed to lie on the circle. The phrase 'cyclic quadrilateral' in a question is a strong hint to use the 180° rule.

请记住，并不是所有画在圆内的四边形都是严格意义上的圆内接四边形，除非四个顶点都被确认在圆周上。题目中如果出现“圆内接四边形”这个词，就是使用 180° 规则的强烈提示。

6. Chord Properties: Perpendicular from Centre | 弦的性质：圆心到弦的垂线

A line drawn from the centre of a circle perpendicular to a chord bisects the chord.

从圆心作弦的垂线，必平分这条弦。

If OM ⊥ AB, then AM = MB

This property is often combined with Pythagoras' theorem to find lengths in circle problems. If you know the radius and the distance from the centre to the chord, you can find half of the chord length.

这个性质常与勾股定理结合，用来求圆中的线段长度。如果你知道半径和圆心到弦的距离，就可以求出弦长的一半。

For example, if the radius is 10 cm and the perpendicular distance from the centre to the chord is 6 cm, then half the chord length is √10² − 6² = √64 = 8 cm, so the full chord is 16 cm.

例如，如果半径为 10 cm，圆心到弦的垂直距离为 6 cm，那么弦长的一半为 √10² − 6² = √64 = 8 cm，因此整条弦长为 16 cm。

Remember that the converse is also true: a line from the centre to the midpoint of a chord is perpendicular to the chord. This can be used to prove right angles in circle questions.

请记住，逆定理也成立：从圆心到弦中点的线段垂直于这条弦。这可以用来证明圆问题中的直角。

7. Tangent and Radius Theorem | 切线与半径定理

A tangent to a circle is perpendicular to the radius drawn to the point of contact.

圆的切线垂直于过切点的半径。

If PA is a tangent and OA is the radius, then ∠OAP = 90°

This is one of the most frequently tested facts in IGCSE geometry. Whenever you see a tangent and a radius meeting, mark the right angle immediately.

这是 IGCSE 几何中最常考的知识点之一。每当你看到切线与半径相交时，请立即标出直角。

This theorem often appears in problems that ask you to find the length of a tangent or the distance from an external point to the centre. The right angle allows you to apply Pythagoras' theorem in

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