Properties of Circles: Radius, Diameter, Angles and Chords | 圆的性质：半径、直径、圆周角与弦

📚 Properties of Circles: Radius, Diameter, Angles and Chords | 圆的性质：半径、直径、圆周角与弦

Circles are one of the most fundamental shapes in geometry, and understanding their properties is essential for Edexcel IGCSE Mathematics. This guide breaks down the key theorems involving radii, diameters, angles, and chords, with clear explanations and worked examples.

圆是几何学中最基本的图形之一，掌握圆的性质对 Edexcel IGCSE 数学考试至关重要。本指南将系统讲解涉及半径、直径、圆周角与弦的核心定理，并提供清晰的例题解析。

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1. Basic Definitions: Radius, Diameter, Chord, and Angle at the Circumference | 基本定义：半径、直径、弦与圆周角

Before exploring theorems, it is important to recall the definitions. The radius is the distance from the centre to any point on the circle (denoted r). The diameter is twice the radius, passing through the centre and connecting two points on the circumference (d = 2r). A chord is any straight line segment whose endpoints lie on the circle. The diameter is the longest chord. An angle at the circumference is an angle formed by two chords that share an endpoint on the circle.

在探索定理之前，先回顾基本定义。半径是从圆心到圆上任意一点的距离，记为 r。直径是半径的两倍，穿过圆心，连接圆上两点（d = 2r）。弦是连接圆上任意两点的线段，直径是最长的弦。圆周角是由两条共端点的弦在圆上形成的角。

• Radius / 半径: centre to circumference (圆心到圆周).
• Diameter / 直径: longest chord, passes through centre (最长的弦，穿过圆心).
• Chord / 弦: any segment joining two points on the circle (连接圆上两点的线段).
• Angle at the circumference / 圆周角: angle with vertex on the circle (顶点在圆上的角).

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2. Theorem 1: Radius Perpendicular to a Chord Bisects It | 定理1：垂直于弦的半径平分弦

If a radius is drawn perpendicular to a chord, it will bisect that chord. In other words, the perpendicular from the centre to a chord divides the chord into two equal parts. This is a powerful tool for finding missing lengths when combined with Pythagoras’ theorem.

如果从圆心作一条垂直于弦的半径，该半径将平分这条弦。换句话说，圆心到弦的垂线把弦分成两条相等的线段。结合勾股定理，这一性质是求未知长度的有力工具。

If OC ⟂ AB, then AC = CB; also OA = OB (radii).

若 OC ⟂ AB，则 AC = CB；且 OA = OB（半径相等）。

Example / 例题: In a circle of radius 10 cm, chord AB is 16 cm long. Find the distance from the centre to the chord.

例题：在半径为 10 cm 的圆中，弦 AB 长 16 cm，求圆心到弦的距离。

Let M be the midpoint of AB. Then AM = 8 cm. In right triangle OMA, OA = 10 cm, AM = 8 cm, so OM = √(10² − 8²) = √36 = 6 cm. The distance is 6 cm.

设 M 为 AB 的中点，则 AM = 8 cm。在直角三角形 OMA 中，OA = 10 cm，AM = 8 cm，因此 OM = √(10² − 8²) = √36 = 6 cm。距离为 6 cm。

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3. Theorem 2: The Angle in a Semicircle is a Right Angle | 定理2：半圆内的圆周角是直角

This is one of the most well-known circle theorems. If a triangle is inscribed in a circle such that one side is the diameter, then the angle opposite that side is always 90°. This theorem links the diameter directly to a right angle at the circumference.

这是最著名的圆定理之一。如果一个三角形内接于圆，且其中一边为直径，那么这条边所对的角恒为 90°。该定理将直径与圆周上的直角直接联系起来。

If AB is a diameter, then ∠ACB = 90° for any point C on the circumference.

若 AB 为直径，则对于圆周上任意点 C，∠ACB = 90°。

Example / 例题: In the diagram, AB is a diameter, and AB = 13 cm, BC = 5 cm. Find AC.

例题：在图中，AB 为直径，AB = 13 cm，BC = 5 cm，求 AC。

Since ∠ACB = 90°, triangle ABC is right-angled at C. Thus AC = √(13² − 5²) = √(169 − 25) = √144 = 12 cm.

因为 ∠ACB = 90°，三角形 ABC 在 C 处为直角三角形，因此 AC = √(13² − 5²) = √(169 − 25) = √144 = 12 cm。

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4. Theorem 3: The Angle at the Centre is Twice the Angle at the Circumference | 定理3：圆心角是圆周角的两倍

For a given arc, the angle subtended at the centre of the circle is exactly twice the angle subtended at any point on the circumference. This theorem is fundamental because it forms the basis for many other circle properties.

对于同一段弧，圆心处所对的角恰好是圆周上任意一点处所对角的两倍。该定理是许多其他圆性质的基础。

∠AOB = 2 × ∠ACB

∠AOB = 2 × ∠ACB

Example / 例题: If ∠ACB = 35°, find ∠AOB.

例题：若 ∠ACB = 35°，求 ∠AOB。

∠AOB = 2 × 35° = 70°.

∠AOB = 2 × 35° = 70°。

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5. Theorem 4: Angles in the Same Segment are Equal | 定理4：同一弧上的圆周角相等

If two angles are subtended by the same chord and lie on the same side of that chord, they are equal. These angles are said to be in the same segment. This theorem is frequently tested in exams, often combined with other angle rules.

如果两个角由同一条弦所对，并且位于该弦的同侧，那么这两个角相等。这样的角被称为在同一弧内。该定理在考试中经常出现，常与其他角度规则结合使用。

∠APB = ∠AQB (if P and Q lie on the same arc AB)

∠APB = ∠AQB（若 P 和 Q 位于同一段弧 AB 上）

Example / 例题: In a cyclic quadrilateral ABCD, ∠ABD = 40° and ∠ACD = x°. Find x.

例题：在圆内接四边形 ABCD 中，∠ABD = 40°，∠ACD = x°，求 x。

Both angles are subtended by chord AD. Thus x = 40°.

两个角都由弦 AD 所对，因此 x = 40°。

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6. Cyclic Quadrilaterals: Opposite Angles Sum to 180° | 圆内接四边形：对角之和为 180°

A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the circumference of a circle. A key property is that opposite angles are supplementary, meaning they add up to 180°. This can be derived directly from the theorem that the angle at the centre is twice the angle at the circumference.

圆内接四边形是四个顶点都在圆上的四边形。其关键性质是对角互补，即对角之和为 180°。这可以直接由圆心角是圆周角两倍的定理推导出来。

∠A + ∠C = 180° and ∠B + ∠D = 180°

∠A + ∠C = 180°，∠B + ∠D = 180°

Example / 例题: In a cyclic quadrilateral, one angle is 75°. What is its opposite angle?

例题：在圆内接四边形中，一个角为 75°，它的对角是多少？

Opposite angle = 180° − 75° = 105°.

对角 = 180° − 75° = 105°。

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7. Tangent to a Circle: Perpendicular to Radius | 圆的切线：与半径垂直

A tangent is a line that touches the circle at exactly one point. The radius drawn to the point of tangency is perpendicular to the tangent. Additionally, from an external point, two tangents drawn to a circle are equal in length. These properties are frequently used in solving geometric problems.

切线是与圆恰好有一个公共点的直线。过切点的半径垂直于切线。此外，从圆外一点引两条切线，它们的长度相等。这些性质常用于解决几何问题。

Radius ⊥ Tangent at the point of contact / 切点处的半径 ⊥ 切线

Example / 例题: A circle has a tangent at point T. The radius OT is 6 cm. If a line from O to an external point P is 10 cm, find PT.

例题：圆在点 T 处有一条切线。半径 OT = 6 cm。若从圆心 O 到外点 P 的距离为 10 cm，求 PT。

Since OT ⟂ PT, triangle OTP is right-angled. Thus PT = √(10² − 6²) = √64 = 8 cm.

因为 OT ⟂ PT，三角形 OTP 为直角三角形，因此 PT = √(10² − 6²) = √64 = 8 cm。

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8. Tangent-Chord Theorem (Alternate Segment Theorem) | 切线-弦定理（弦切角定理）

The tangent-chord theorem states that the angle between a tangent and a chord drawn from the point of contact is equal to the angle in the alternate segment. This theorem is also known as the alternate segment theorem and is a powerful tool for finding unknown angles.

切线-弦定理指出，切线与过切点所作的弦之间的夹角，等于该弦所对的另一段弧上的圆周角。这一定理也被称为“弦切角定理”，是求未知角的有力工具。

Angle between tangent and chord = angle in the alternate segment

切线与弦的夹角 = 另一段弧上的圆周角

Example / 例题: A tangent at point A and chord AB form an angle of 50°. What is the angle in the alternate segment?

例题：过点 A 的切线与弦 AB 形成 50° 的角，求另一段弧上的圆周角。

The angle in the alternate segment is equal to 50°.

另一段弧上的圆周角等于 50°。

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9. Intersecting Chords Theorem | 相交弦定理

When two chords intersect inside a circle, the product of the lengths of the segments of one chord equals the product of the segments of the other chord. This is known as the intersecting chords theorem. It provides a direct method for finding unknown lengths without needing angle information.

当两条弦在圆内相交时，一条弦被分成的两段长度之积等于另一条弦被分成的两段长度之积。这就是相交弦定理。它为求未知长度提供了快捷方法。

If chords AB and CD intersect at P, then PA × PB = PC × PD.

若弦 AB 与 CD 相交于点 P，则 PA × PB = PC × PD。

Example / 例题: Chords AB and CD intersect at P. PA = 4 cm, PB = 6 cm, PC = 3 cm. Find PD.

例题：弦 AB 与 CD 相交于点 P。PA = 4 cm，PB = 6 cm，PC = 3 cm，求 PD。

4 × 6 = 3 × PD, so PD = 24 ÷ 3 = 8 cm.

4 × 6 = 3 × PD，因此 PD = 24 ÷ 3 = 8 cm。

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10. Worked Example: Combining Theorems | 综合例题：多定理结合

The following example demonstrates how multiple circle theorems can be applied together in one problem. Effective problem-solving requires identifying which theorems apply to which parts of the diagram.

下面的例题展示了如何在同一个问题中综合运用多个圆定理。有效的解题关键在于识别图形中各个部分适用的定理。

In the diagram, O is the centre, AB is a diameter, and C is a point on the circumference. Given that ∠OCB = 25°, find ∠CAB and ∠ACB.

在图中，O 为圆心，AB 为直径，C 为圆周上一点。已知 ∠OCB = 25°，求 ∠CAB 和 ∠ACB。

Since AB is a diameter, ∠ACB = 90° (angle in a semicircle). Triangle OCB is isosceles because OC = OB (both radii). Therefore ∠OBC = ∠OCB = 25°. In triangle ABC, ∠CAB = 180° − 90° − 25° = 65°.

因为 AB 为直径，所以 ∠ACB = 90°（半圆内的圆周角为直角）。三角形 OCB 为等腰三角形，因为 OC = OB（都是半径），因此 ∠OBC = ∠OCB = 25°。在三角形 ABC 中，∠CAB = 180° − 90° − 25° = 65°。

Answer / 答案: ∠ACB = 90° (right angle in a semicircle) and ∠CAB = 65°.

答案：∠ACB = 90°（半圆内的直角），∠CAB = 65°。

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11. Summary of Key Theorems | 关键定理汇总

The table below summarises the essential circle theorems covered in this article. Keep this as a revision reference when solving Edexcel IGCSE geometry problems.

下表汇总了本文涉及的核心圆定理。做题时可将此表作为复习参考。

Theorem / 定理	Statement / 描述
Radius-Chord / 半径与弦	Radius perpendicular to a chord bisects it / 垂直于弦的半径平分弦
Semicircle / 半圆	Angle in a semicircle is 90° / 半圆内的圆周角为 90°
Centre-Circumference / 圆心角与圆周角	Angle at centre = 2 × angle at circumference / 圆心角 = 2 × 圆周角
Same Segment / 同弧	Angles in the same segment are equal / 同一弧上的圆周角相等
Cyclic Quadrilateral / 圆内接四边形	Opposite angles sum to 180° / 对角之和为 180°
Tangent / 切线	Radius perpendicular to tangent / 半径垂直于切线
Alternate Segment / 弦切角	Tangent-chord angle = angle in alternate segment / 切线弦夹角等于另一段弧上的圆周角
Intersecting Chords / 相交弦	PA × PB = PC × PD / PA × PB = PC × PD

Mastering these circle theorems will allow you to tackle a wide range of IGCSE Mathematics questions with confidence. Remember to always look for radii, diameters, tangents, and cyclic quadrilaterals when analysing a diagram.

熟练掌握这些圆定理，你就能自信应对 IGCSE 数学中各类相关的题目。请记住，在分析图形时，务必留意半径、直径、切线和圆内接四边形等关键特征。

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