📚 IGCSE Maths: Circle Theorems | IGCSE 数学：圆定理
Circle theorems are a core part of IGCSE Mathematics. They describe the relationships between angles, chords, tangents and radii in a circle. Students need to recognise the diagrams, state the correct theorem and use algebra to find unknown angles.
圆定理是 IGCSE 数学的核心内容。它们描述了圆中角、弦、切线和半径之间的关系。学生需要识别图形、说出正确的定理，并运用代数求出未知角度。
1. Key Circle Terms | 圆的基本术语
Before using circle theorems, you must know the language of circles. The most common terms are radius, diameter, chord, tangent, arc, sector, segment and circumference. A clear diagram usually shows these parts.
在使用圆定理之前，你必须掌握圆的基本术语。最常见的术语有半径、直径、弦、切线、弧、扇形、弓形和圆周。清晰的图形通常会标出这些部分。
| Radius | 半径 | A line from the centre to the circumference. |
| Chord | 弦 | A line joining two points on the circumference. |
| Tangent | 切线 | A line touching the circle at exactly one point. |
| Arc | 弧 | Part of the circumference of a circle. |
2. Angle at the Centre Theorem | 圆心角定理
The angle at the centre of a circle is twice the angle at the circumference when both angles are subtended by the same arc. This is often the first theorem used in multi-step IGCSE questions.
当两个角都对着同一条弧时，圆心角是圆周角的两倍。这通常是 IGCSE 多步计算题中第一个用到的定理。
∠AOB = 2 × ∠APB
In the diagram, if arc AB subtends angle AOB at the centre and angle APB at the circumference, then the centre angle AOB is double the circumference angle APB.
在图中，如果弧 AB 在圆心处对着角 AOB，在圆周上对着角 APB，那么圆心角 AOB 是圆周角 APB 的两倍。
3. 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. This theorem is useful because it creates a right-angled triangle, allowing Pythagoras' theorem or trigonometry to be used.
直径在圆周上所对的角总是直角。这个定理很有用，因为它构造了直角三角形，从而可以使用勾股定理或三角学。
∠APB = 90°
If AB is a diameter and P is any point on the circumference, then angle APB is exactly 90°. Students should look for diameter lines to spot this theorem.
如果 AB 是直径，P 是圆周上的任意一点，那么角 APB 恰好是 90°。学生应注意寻找直径线来识别这个定理。
4. Angles in the Same Segment | 同弓形内的圆周角
Angles subtended by the same chord in the same segment of a circle are equal. The chord creates two segments, and the angles in the same segment face the same arc.
同一条弦在同一弓形内所对的圆周角相等。弦将圆分成两个弓形，同一弓形内的角面对同一条弧。
∠APB = ∠AQB
If points P and Q lie on the same side of chord AB, then angles APB and AQB are equal. This theorem helps to transfer an angle from one point to another.
如果点 P 和 Q 在弦 AB 的同一侧，那么角 APB 和角 AQB 相等。这个定理有助于将一个角从一点转移到另一点。
5. Cyclic Quadrilateral Theorem | 圆内接四边形定理
A cyclic quadrilateral is a four-sided shape with all vertices on the circumference of a 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圆周上的四边形。圆内接四边形的对角之和为 180°。
∠A + ∠C = 180° and ∠B + ∠D = 180°
This theorem works both ways: if opposite angles sum to 180°, the four points lie on a circle. IGCSE questions often ask you to prove that a quadrilateral is cyclic.
这个定理反过来也成立：如果对角之和为 180°，那么四个点共圆。IGCSE 题目经常要求证明一个四边形是圆内接四边形。
6. Tangent and Radius | 切线与半径
A tangent to a circle is perpendicular to the radius at the point of contact. This is the most basic tangent theorem and is used in many right-angle calculations.
圆的切线与经过切点的半径垂直。这是最基本的切线定理，广泛用于直角计算中。
OT ⟂ PT
If T is the point where tangent PT touches the circle and O is the centre, then radius OT is perpendicular to tangent PT. The right angle is a key clue in diagrams.
如果 T 是切线 PT 与圆的切点，O 是圆心，那么半径 OT 与切线 PT 垂直。图中的直角是重要的提示。
7. Tangents from an External Point | 从外点引出的切线
Two tangents drawn from the same external point to a circle are equal in length. This theorem creates congruent triangles and is often linked to the tangent-radius theorem.
从同一个外部点向圆引出的两条切线长度相等。这个定理构造了全等三角形，并且经常与切线-半径定理结合使用。
PA = PB
If PA and PB are tangents from external point P to the circle, then PA equals PB. The line joining P to the centre bisects the angle between the tangents.
如果 PA 和 PB 是从外部点 P 向圆引出的两条切线，那么 PA 等于 PB。连接 P 和圆心的直线平分两条切线之间的夹角。
8. Alternate Segment Theorem | 交替弓形定理
The angle between a tangent and a chord is equal to the angle in the alternate segment. This theorem is considered one of the more demanding circle theorems because students must identify the alternate segment correctly.
切线与弦之间的夹角等于交替弓形内的圆周角。这个定理被认为是较难的圆定理之一，因为学生必须正确识别交替弓形。
∠TPB = ∠PAB
If PT is a tangent and PB is a chord, then angle TPB equals angle PAB, where A is any point on the circle in the alternate segment. This theorem often appears in proof questions.
如果 PT 是切线，PB 是弦，那么角 TPB 等于角 PAB，其中 A 是交替弓形内圆周上的任意一点。这个定理经常出现在证明题中。
9. Perpendicular from Centre to Chord | 圆心到弦的垂线
A perpendicular drawn from the centre of a circle to a chord bisects the chord. This theorem is useful for finding lengths and shows the symmetry of circles.
从圆心向弦所作的垂线平分该弦。这个定理有助于求线段长度，也体现了圆的对称性。
OD ⟂ AB 且 AD = DB
If OD is perpendicular to chord AB, then D is the midpoint of AB. Conversely, the line joining the centre to the midpoint of a chord is perpendicular to the chord.
如果 OD 垂直于弦 AB，那么 D 是 AB 的中点。反过来，连接圆心与弦中点的直线垂直于弦。
10. Worked Example | 例题解析
Worked examples help students see how several circle theorems can be used together. In an IGCSE question, you rarely use one theorem on its own.
例题解析有助于学生理解如何综合运用多个圆定理。在 IGCSE 考试题中，很少单独使用一个定理。
on the circumference. AB is a diameter, and angle CAB = 32°. Find angle ABC and angle ACB.
一个圆的圆心为 O。点 A、B、C 和 D 在圆周上。AB 是直径，角 CAB = 32°。求角 ABC 和角 ACB。
heorem. In triangle ABC, the sum of angles is 180°, so angle ABC = 180° - 90° - 32° = 58°.
因为 AB 是直径，根据半圆上的圆周角定理，角 ACB 为 90°。在三角形 ABC 中，内角和为 180°，所以角 ABC = 180° - 90° - 32° = 58°。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Students often confuse the angle at the centre theorem with the angles in the same segment theorem. Always check where the vertex of each angle sits before choosing a theorem.
学生经常混淆圆心角定理与同弓形内的圆周角定理。在选择定理之前，务必检查每个角的顶点位置。
- Label every point in the diagram before starting. | 开始前先在图中标出每个点。
- Write the theorem next to each step, such as 'angle in semicircle = 90°'. | 每一步旁边写出定理，例如“半圆上的圆周角 = 90°”。
- Check for diameters and tangents first, because they give right angles. | 首先检查直径和切线，因为它们能提供直角。
- Use algebra when angles are given as expressions, such as 2x + 10°. | 当角度以表达式给出时，例如 2x + 10°，要使用代数。
12. Summary of Circle Theorems | 圆定理总结
The eight main circle theorems cover angle at centre, angle in semicircle, same segment, cyclic quadrilateral, tangent-radius, tangents from a point, alternate segment and chord bisection. Knowing these will cover most IGCSE geometry questions.
八个主要圆定理包括圆心角、半圆上的圆周角、同弓形内的圆周角、圆内接四边形、切线-半径、从一点引出的切线、交替弓形以及弦的平分。掌握这些定理就能应对大多数 IGCSE 几何题。
∠AOB = 2∠APB, ∠APB = 90°, ∠APB = ∠AQB, ∠A + ∠C = 180°, OT ⟂ PT, PA = PB, ∠TPB = ∠PAB, AD = DB
Practise drawing circle diagrams from memory and labelling the relevant arc, chord or tangent. This visual recall is essential for exam success.
练习凭记忆画出圆的各种图形，并标出相关的弧、弦或切线。这种视觉记忆对考试成功至关重要。
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