Circle Theorems (T-3-1086) | 圆定理（T-3-1086）

📚 Circle Theorems (T-3-1086) | 圆定理（T-3-1086）

Circle theorems are a fundamental part of the IGCSE Mathematics syllabus, especially in the Geometry and Measures section. They allow you to solve angle problems without direct measurement, using logical deduction. This article follows the TutorHao revision plan for topic T-3-1086, covering every key theorem, worked examples, and exam tips.

圆定理是 IGCSE 数学教学大纲中的一个基础板块，尤其在几何与测量部分。它们让你能够不借助直接测量，而通过逻辑推理来解决角度问题。本文遵循 TutorHao 针对 T-3-1086 主题的复习计划，涵盖所有关键定理、例题和应试技巧。

1. What Are Circle Theorems? | 什么是圆定理？

A circle theorem is a mathematical statement about a circle that is always true. These statements relate angles, chords, tangents, and radii in predictable ways, which makes it possible to find unknown values precisely.

圆定理是关于圆的一个数学陈述，它始终成立。这些陈述以可预测的方式将角度、弦、切线和半径联系起来，从而让我们能够精确地求出未知量。

In the IGCSE examination, you may be asked to state a theorem, calculate an angle, or prove why a result is true. A strong understanding of each theorem and its diagram is therefore essential for high marks.

在 IGCSE 考试中，你可能会被要求陈述一个定理、计算某个角，或证明某个结果为什么成立。因此，深刻理解每个定理及其图形，是获得高分的关键。

2. Key Circle Vocabulary | 圆的基本概念

Before you study the theorems, you must be comfortable with the language of circles. The radius is the distance from the centre to the circumference; a chord is a straight line joining two points on the circle; a tangent is a line that touches the circle at exactly one point; an arc is part of the circumference; a sector is the region between two radii and an arc; and a segment is the region between a chord and the arc it cuts off.

在学习定理之前，你必须熟悉圆的术语。半径是从圆心到圆周的距离；弦是连接圆上两点的线段；切线是与圆恰好相交于一点的直线；弧是圆周的一部分；扇形是两条半径与一段弧围成的区域；弓形是一条弦与其所截弧围成的区域。

The diameter is the longest chord in a circle and passes through the centre. It is always twice the radius. These basic terms appear in almost every circle theorem problem.

直径是圆中最长的弦，并且穿过圆心。它总是半径的两倍。这些基础术语几乎出现在所有圆定理题目中。

3. Theorem 1: Angle at the Centre | 定理1：圆心角是圆周角的两倍

This theorem states that the angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the circumference.

该定理指出：同一段弧在圆心处所对的角，等于同一段弧在圆周上任意一点所对的角的两倍。

∠AOB = 2 × ∠ACB

Here O is the centre, A and B are points on the circumference, and C is another point on the circumference. For example, if ∠AOB = 80°, then ∠ACB = 40°.

这里 O 是圆心，A 和 B 是圆周上的点，C 是圆周上的另一点。例如，如果 ∠AOB = 80°，那么 ∠ACB = 40°。

4. Theorem 2: Angle in a Semicircle | 定理2：半圆内的圆周角为直角

If a triangle is inscribed in a circle with one side as the diameter, then the angle opposite that diameter is always 90°. This is a direct consequence of Theorem 1 because the angle at the centre is 180°.

如果一个三角形内接于圆，且一条边是直径，那么该直径所对的角总是 90°。这是定理1的直接推论，因为此时的圆心角等于 180°。

If AB is a diameter, then ∠ACB = 90°

This theorem is particularly useful in combined shapes, such as a triangle inside a semicircle or a rectangle inscribed in a circle.

这个定理在组合图形中特别有用，例如半圆内的三角形或内接于圆的矩形。

5. Theorem 3: Angles in the Same Segment | 定理3：同弧上的圆周角相等

Angles subtended by the same chord and on the same side of the chord are equal. In other words, if several points lie on the same arc, they all give the same angle for a fixed chord.

同一弦所对的、并且在弦同侧的圆周角相等。换句话说，如果多个点位于同一段弧上，那么它们对同一固定弦所成的角都相等。

∠ADB = ∠ACB

This theorem is often used when two triangles share the same base inside a circle, as it immediately gives equal angles without further calculation.

当两个三角形在圆内共用底边时，这个定理可立即得出相等角，无需进一步计算。

6. Theorem 4: Cyclic Quadrilateral | 定理4：圆内接四边形对角互补

A cyclic quadrilateral is a four-sided figure whose vertices all lie on one circle. The opposite angles of a cyclic quadrilateral add up to 180°.

圆内接四边形是四个顶点都在同一个圆上的四边形。圆内接四边形的对角之和等于 180°。

∠A + ∠C = 180°, ∠B + ∠D = 180°

For example, if ∠A = 110°, then ∠C = 70°. This rule can also be used to prove whether a quadrilateral is cyclic.

例如，若 ∠A = 110°，则 ∠C = 70°。该规则也可用于判断一个四边形是否内接于圆。

7. Theorem 5: Tangent and Radius | 定理5：切线与半径垂直

A tangent to a circle is perpendicular to the radius at the point of contact. Therefore, the angle between the tangent and the radius is exactly 90°.

圆的切线在切点处与过该点的半径互相垂直。因此，切线与半径的夹角恰好是 90°。

Radius ⊥ Tangent

This theorem is often the first step in tangent problems. Always draw the radius to the point of contact and label the right angle clearly.

在切线问题中，这通常是第一步。务必画出切点处的半径，并清楚标出直角。

8. Theorem 6: Alternate Segment Theorem | 定理6：切线-弦定理

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. This is also known as the tangent-chord theorem.

切线与通过切点的弦之间的夹角，等于弦所对的、位于交替段的圆周角。这个定理也叫弦切角定理。

Angle(tangent, chord) = angle in the alternate segment

For example, if a tangent at A and chord AB form a 60° angle, then the angle in the alternate segment subtended by AB is also 60°.

例如，若在 A 点的切线与弦 AB 成 60° 角，则 AB 所对的交替段圆周角也等于 60°。Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

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Circle Theorems (T-3-1086) | 圆定理（T-3-1086）