IGCSE Circle Theorems | IGCSE 圆的定理

📚 IGCSE Circle Theorems | IGCSE 圆的定理

Circle theorems are one of the most rewarding topics in IGCSE Mathematics. Once you understand the geometric relationships hidden inside a circle, a whole family of exam questions becomes simple and predictable. This guide covers every theorem you need, with clear explanations and proven strategies to solve even the most challenging problems.

圆定理是 IGCSE 数学中回报率最高的内容之一。一旦你理解了隐藏在圆内部的几何关系，大量考试题目都会变得简单而可预测。本指南涵盖你所需的全部定理，提供清晰的解释和经过验证的解题策略，帮助你应对最具挑战性的问题。

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1. Circle Basics | 圆的基础概念

rcumference, so d = 2r. A chord is any straight line joining two points on the circumference. A tangent is a line that touches the circle at exactly one point. An arc is part of the circumference, and a sector is the region between two radii and an arc.

在学习定理之前，你必须熟练掌握圆的术语。圆心是圆内固定的点，标记为 O。半径 (r) 是从圆心到圆周上任意一点的距离。直径 (d) 穿过圆心并连接圆周上的两点，因此 d = 2r。弦是连接圆周上两点的任意线段。切线是在恰好一个点上接触圆的直线。弧是圆周的一部分，扇形是两条半径与一段弧之间的区域。

The table below summarises the key terms you will meet in this topic:

下表总结了本话题中你会遇到的关键术语：

Term	Meaning
Radius (r)	Centre to circumference / 圆心到圆周
Diameter (d)	2 × radius / 半径的两倍
Chord	Line joining two circumference points / 连接圆周两点的线段
Tangent	Line touching the circle once / 与圆只有一个交点的直线
Arc	Part of the circumference / 圆周的一部分
Segment	Region between chord and arc / 弦与弧之间的区域

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2. Theorem 1: Angle in a Semicircle | 定理1：半圆内的圆周角

If a triangle is drawn inside a circle so that one of its sides is the diameter, then the angle opposite that diameter is always a right angle (90°).

如果一个三角形内接于圆，且它的一条边是直径，那么这条直径所对的角始终是直角（90°）。

∠ACB = 90°

For example, if A and B are the endpoints of a diameter and C is any point on the circumference, then triangle ACB is right-angled at C. No matter where C is placed, the angle stays at 90°. This theorem is often combined with Pythagoras' theorem to find missing lengths.

例如，如果 A 和 B 是直径的端点，C 是圆周上任意一点，那么三角形 ACB 在 C 处为直角三角形。无论 C 在何处，这个角始终保持为 90°。该定理经常与勾股定理结合来求缺失的长度。

Exam tip: whenever you spot a diameter in a circle question, immediately look for the right angle it creates on the circumference. This single observation often unlocks the whole problem.

考试提示：当你在圆的问题中发现直径时，立刻寻找它在圆周上形成的直角。这一个观察往往能解开整道题。

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3. Theorem 2: Angle at the Centre | 定理2：圆心角与圆周角

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.

同一条弧在圆心处的夹角，等于它在圆周上任意一点处的夹角的两倍。

∠AOB = 2 × ∠ACB

nd A, B and C all lie on the circumference. The arc AB is the same arc for both angles. If ∠AOB = 80°, then ∠ACB = 40°; conversely, if ∠ACB = 35°, then ∠AOB = 70°. Remember that the two angles must stand on the same arc, not on different arcs.

这里 O 是圆心，A、B、C 都位于圆周上。对于两个角来说，弧 AB 是同一条弧。如果 ∠AOB = 80°，那么 ∠ACB = 40°；反过来，如果 ∠ACB = 35°，那么 ∠AOB = 70°。请记住，两个角必须对应同一条弧，而不是不同的弧。

This theorem is the most frequently tested circle theorem in IGCSE papers, so practise applying it in both directions.

该定理是 IGCSE 试卷中最常考查的圆定理，因此务必练习从两个方向应用它。

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4. 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 at the circumference are equal, provided the vertices lie on the same side of the chord.

在圆周上，由同一条弦所夹的角相等，前提是角的顶点位于弦的同一侧。

∠ACB = ∠ADB

In this scenario, C and D are both points on the circumference, and both angles stand on chord AB. Because C and D are on the same side of AB, the angles are equal. This is useful when a diagram contains several points on the same arc and you need to transfer an angle from one position to another.

在这种情况下，C 和 D 都是圆周上的点，且两个角都对应弦 AB。由于 C 和 D 位于 AB 的同一侧，所以两个角相等。当图中同一段弧上有多个点，且你需要在不同位置之间转移角度时，这条定理非常有用。

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5. Theorem 4: Cyclic Quadrilateral | 定理4：圆内接四边形

Opposite angles of a cyclic quadrilateral sum to 180°. A cyclic quadrilateral is a four-sided shape whose four vertices all lie on the circumference of a circle.

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

∠a + ∠c = 180°, ∠b + ∠d = 180°

This property is unique to cyclic quadrilaterals and does not hold for ordinary quadrilaterals. If you know three of the four angles in a cyclic quadrilateral, you can always find the fourth by subtracting the opposite angle from 180°.

这一性质是圆内接四边形独有的，不适用于普通四边形。如果你已知圆内接四边形的三个角，你总能用 180° 减去它的对角来求出第四个角。

Worked example: if ∠a = 70° and ∠b = 110°, then ∠c = 180° − 70° = 110° and ∠ 180° − 110° = 70°. The quadrilateral in this case is a parallelogram-shaped cyclic figure.

示例：如果 ∠a = 70° 且 ∠b = 110°，那么 ∠c = 180° − 70° = 110°，∠d = 180° − 110° = 70°。此时

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