G-2 Geometry: Circle Theorems and Angle Rules in IGCSE Mathematics | G-2 几何：圆定理与角度规则（IGCSE 数学）

📚 G-2 Geometry: Circle Theorems and Angle Rules in IGCSE Mathematics | G-2 几何：圆定理与角度规则（IGCSE 数学）

This article follows the structure of the G-2 section in the TutorHao teacher’s guide. It focuses on the angle rules and circle theorems that are frequently tested in IGCSE Mathematics extended papers. We will explain each theorem, show the standard naming conventions, and include worked examples to help you apply them confidently.

本文依据 TutorHao 教师用书中 G-2 部分的顺序编写，聚焦 IGCSE 数学扩展卷中高频考查的角度规则与圆定理。我们将逐一讲解定理、规范几何语言，并通过例题帮助大家熟练运用。

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1. Fundamental Angle Rules | 基础角度规则

Before studying circle theorems, you must be comfortable with the basic angle rules. Angles on a straight line add to 180°, angles around a point add to 360°, vertically opposite angles are equal, and the interior angles of a triangle add to 180°. These rules are the building blocks for all later proofs.

在学习圆定理之前，必须熟练基础角度规则：同一直线上的角之和为 180°，一个点周围的角之和为 360°，对顶角相等，三角形内角之和为 180°。这些规则是后续所有证明的基础。

• Adjacent angles on a straight line: a + b = 180°. | 同一直线上的相邻角：a + b = 180°。

• Angles around a point: a + b + c + d = 360°. | 一个点周围的角：a + b + c + d = 360°。

• Vertically opposite angles are equal. | 对顶角相等。

• Corresponding angles, alternate angles and co-interior angles are formed by parallel lines. | 平行线产生同位角、内错角与同旁内角。

Angles on a straight line: a + b = 180° | 同一直线上的角：a + b = 180°

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2. Basic Terms of a Circle | 圆的基本术语

A clear understanding of circle vocabulary is essential for identifying which theorem to use. The circumference is the outer boundary of the circle. A radius is a line from the centre to the circumference. A diameter is a straight line passing through the centre and touching the circumference at two points. A chord is a line segment joining two points on the circumference. A tangent is a line that touches the circle at exactly one point.

牢固掌握圆的基本术语是正确选择定理的前提。圆周（circumference）是圆的外边界；半径（radius）是圆心到圆周的线段；直径（diameter）是经过圆心并连接圆周两点的线段；弦（chord）是连接圆周上两点的线段；切线（tangent）是与圆只有一个公共点的直线。

Term | 术语	Meaning | 含义
Arc | 弧	Part of the circumference | 圆周的一部分
Sector | 扇形	Region bounded by two radii and an arc | 两条半径与一段弧围成的区域
Segment | 弓形	Region bounded by a chord and an arc | 一段弦与一段弧围成的区域
Centre | 圆心	The fixed point inside the circle | 圆内部固定不变的点

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3. Chord Properties | 弦的性质

The perpendicular from the centre of a circle to a chord bisects the chord. This means that if a radius is drawn at right angles to a chord, it cuts the chord into two equal parts. The converse is also true: the line from the centre which bisects a chord is perpendicular to the chord.

圆心到弦的垂线平分这条弦。也就是说，如果一条半径垂直于弦，那么它一定把弦分成两段相等部分。反过来也成立：圆心连到弦中点的直线必定垂直于弦。

If OM ⊥ AB, then AM = MB | 若 OM ⊥ AB，则 AM = MB

All chords that are equal in length are equidistant from the centre. Therefore, if two chords are the same length, their distances from the centre are equal, and the corresponding arcs subtended by those chords are also equal.

等长的弦到圆心的距离相等。因此，如果两条弦相等，它们到圆心的距离相等，并且它们所对的弧也相等。

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4. Angle at the Centre and Circumference | 圆心角与圆周角

The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc. In other words, if two radii form an angle at the centre, and that same arc is viewed from a point on the circumference, the angle at the centre is double the angle at the circumference.

同一条弧所对的圆心角等于它所对的圆周角的两倍。换句话说，如果两条半径在圆心处形成一个角，同一个弧在圆周某一点上看到的角，圆心角是圆周角的两倍。

Angle at centre = 2 × Angle at circumference | 圆心角 = 2 × 圆周角

This theorem applies only when the two angles are subtended by the same arc or the same chord. Mark the subtended arc clearly in the diagram to avoid using the wrong arc.

这个定理仅在两个角对应同一条弧或同一条弦时成立。在图中清楚标出对应的弧，避免选错弧。

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5. Angle in a Semicircle | 半圆内的角

If a triangle is drawn inside a circle so that one side is the diameter of the circle, the angle opposite this diameter is always a right angle. This is called the angle in a semicircle theorem, sometimes known as Thales’ theorem.

如果三角形内接于一个圆，且一边是圆的直径，那么这条直径所对的角一定是直角。这就是“半圆内的角是直角”定理，也被称为泰勒斯定理。

If AB is the diameter, then ∠ACB = 90° | 若 AB 是直径，则 ∠ACB = 90°

This is one of the most commonly tested theorems in IGCSE papers. When you see a right angle, always check whether the hypotenuse is the diameter of the circumscribed circle.

这是 IGCSE 考试中最常考查的定理之一。看到直角时，应该立即检查直角三角形的斜边是否是外接圆的直径。

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6. Angles in the Same Segment | 同弧上的圆周角

Angles subtended by the same chord at the circumference on the same side of the chord are equal. For example, if points A, B, C and D lie on a circle, then all angles subtended by chord AB at points C and D on the same arc are equal.

同一条弦在圆周上同一侧所对的圆周角相等。例如，点 A、B、C、D 共圆时，弦 AB 在 C、D 两点所对的圆周角相等。

∠ACB = ∠ADB because both subtend chord AB | ∠ACB = ∠ADB，因为它们对应弦 AB

Be careful: C and D must lie in the same segment. If they are on opposite sides of chord AB, the angles are supplementary instead of equal.

注意：C、D 必须处于同一弓形内。如果它们在弦 AB 的两侧，那么两个角的关系是互补而不是相等。

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7. Cyclic Quadrilaterals | 圆内接四边形

A cyclic quadrilateral is a four-sided figure whose vertices all lie on the circumference of a circle. In any cyclic quadrilateral, the sum of each pair of opposite angles is 180°. This is because the opposite angles are subtended by complementary arcs.

圆内接四边形是四个顶点都在同一个圆上的四边形。任何圆内接四边形中，每一对对角的和都是 180°。这是因为这两个角所对的弧互为补弧。

∠A + ∠C = 180° and ∠B + ∠D = 180° | ∠A + ∠C = 180° 且 ∠B + ∠D = 180°

The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. This property is very useful in multi-step geometry questions and is often applied with parallel line angle rules.

圆内接四边形的一个外角等于它的内对角。这一性质在多步几何题中非常实用，经常与平行线角度规则结合使用。

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8. Tangent–Radius Theorem | 切线与半径定理

A tangent to a circle is perpendicular to the radius at the point of contact. If a line touches the circle at point T, then the radius OT is perpendicular to the tangent line at T. This theorem is used to establish right angles in many circle problems.

圆的切线在切点处与过切点的半径垂直。如果一条直线在点 T 与圆相切，那么半径 OT 垂直于点 T 处的切线。该定理用于在圆相关题目中寻找直角。

Radius from centre to tangent point is perpendicular to the tangent | 圆心到切点的半径垂直于切线

Tangents drawn from an external point to a circle are equal in length. Therefore, if two tangents from a point P touch the circle at A and B, then PA = PB. This equal-length property is extremely common in IGCSE extended questions.

从圆外一点引圆的两条切线，切线长相等。因此，如果从点 P 引两条切线分别切圆于 A、B，则 PA = PB。切线长相等的性质在 IGCSE 扩展题中非常常见。

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9. Alternate Segment Theorem | 弦切角定理

The angle between a tangent and a chord drawn from the point of contact is equal to the angle in the alternate segment. If the tangent at point B meets chord AB, then the angle between the tangent and AB equals the angle subtended by AB in the opposite arc.

切线与经过切点的弦所夹的角，等于该弦所对的同侧圆周角，即弦切角定理。若在切点 B 处作弦 AB，则切线与 AB 的夹角等于 AB 在另一侧弧上所对的圆周角。

Angle between tangent and chord = angle in the alternate segment | 切线与弦的夹角 = 弦所对的另一侧圆周角

This theorem is often considered one of the hardest circle theorems because students forget to identify the alternate segment correctly. To avoid mistakes, shade the segment inside the angle between the tangent and chord; the angle at any point on the opposite segment is the one you need.

弦切角定理常被认为是最难的圆定理之一，因为学生难以识别正确的位置。为了避免错误，可以把切线与弦所夹的内部区域涂上阴影，阴影对侧的圆周角就是要求的角。

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10. Intersecting Chords and Secants | 相交弦与割线（拓展）

For two chords AB and CD intersecting inside a circle at point P, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord: PA × PB = PC × PD. This is known as the intersecting chord theorem.

圆内两条弦 AB 和 CD 相交于点 P 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：PA × PB = PC × PD。这就是相交弦定理。

PA × PB = PC × PD | PA × PB = PC × PD

If an external point P has a secant meeting the circle at A and B, and a tangent PC touching the circle at C, then PC² = PA × PB. This is the tangent-secant power theorem. It offers a direct way to find unknown lengths in circle geometry.

如果圆外一点 P 有割线与圆交于 A、B，且切线 PC 与圆切于 C，那么 PC² = PA × PB。这就是切割线定理。它能直接求解圆几何中的未知线段长度。

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11. Worked Examples | 例题精解

Example 1: In the diagram, O is the centre of the circle. The angle at the circumference ∠ABC = 35°. Find the angle at the centre ∠AOC.

例 1：如图，O 是圆心，∠ABC = 35°，求圆心角 ∠AOC。

Using the theorem that the angle at the centre is twice the angle at the circumference subtended by the same arc, we have ∠AOC = 2 × 35° = 70°. Therefore, ∠AOC = 70°.

由同弧所对圆心角等于圆周角的两倍，∠AOC = 2 × 35° = 70°。因此，∠AOC = 70°。

Example 2: A cyclic quadrilateral ABCD has ∠A = 4x and ∠C = 5x. Find the value of x.

例 2：圆内接四边形 ABCD 中，∠A = 4x，∠C = 5x，求 x 的值。

Since the opposite angles of a cyclic quadrilateral are supplementary, ∠A + ∠C = 180°. Therefore, 4x + 5x = 180°, which gives 9x = 180°, so x = 20°.

圆内接四边形对角互补，所以 ∠A + ∠C = 180°。由此可得 4x + 5x = 180°，即 9x = 180°，因此 x = 20°。

Example 3: PQ is a tangent at point T to a circle with centre O. If ∠OTQ = 90° and ∠PTQ = 50°, explain whether the figure is consistent with tangent properties.

例 3：PQ 是圆在点 T 处的切线，圆心为 O。若 ∠OTQ = 90°，∠PTQ = 50°，判断该图形是否满足切线性质。

If T is the point of contact, then OT is the radius and PQ is the tangent. The radius must be perpendicular to the tangent at the point of contact, so ∠OTQ should be 90°. The value 50° is used for a different angle in the same triangle, and the figure can still be consistent if the points are labelled correctly.

若 T 是切点，则 OT 是半径，PQ 是切线。半径与切线的夹角必须是 90°，所以 ∠OTQ 应为 90°。50° 是三角形中另一个角的度数；只要点的标记正确，图形仍然可以满足切线性质。

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12. Common Mistakes and Revision Tips | 常见错误与复习建议

Common errors include applying the angle at the centre theorem to the wrong arc, confusing the alternate segment theorem with the same-side angle theorem, and forgetting to mention the name of the theorem in written geometric proofs. Another frequent mistake is using chord lengths instead of angle measures when applying circle theorems.

常见错误包括：把圆心角定理应用到错误弧上；混用弦切角定理与同弧圆周角定理；在几何证明中忘记写出定理名称；以及在应用圆定理时误把弦长当作角度。

• Always name the theorem using the exact IGCSE wording. | 始终使用 IGCSE 标准表述写出定理名称。

• Draw a temporary radius or chord to find hidden right angles. | 画辅助半径或弦来寻找隐藏的直角。

• Use a diagram and mark every given angle before solving. | 先在图上标出所有已知角，再开始解题。

• Memorise the key relationships: centre angle = 2 × circumference angle; opposite angles in cyclic quadrilateral sum to 180°. | 牢记核心关系：圆心角 = 2 × 圆周角；圆内接四边形对角和为 180°。

Practising with at least ten past-paper questions is the most effective way to master the G-2 section. The more varied the diagrams, the easier it becomes to recognise the theorem hidden inside each question.

解决至少十道历年真题是掌握 G-2 部分最有效的方式。图形变化越丰富，考试中就越容易识别出每道题背后隐藏的定理。

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Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

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