📚 IGCSE Maths G4: Circle Theorems and Angle Reasoning | IGCSE 数学 G4：圆定理与角度推理
Circle theorems are a core part of IGCSE Geometry. They link angles, lengths, tangents, chords and arcs in a circle, and they appear in both structured questions and multi-step proofs. A strong understanding of these theorems will help you justify every step clearly and avoid losing marks in 'show that' questions.
圆定理是 IGCSE 几何的核心内容。它们把圆中的角、长度、切线、弦和弧联系起来，常见于结构化试题和多步证明中。扎实掌握这些定理有助于你清晰地说明每一步理由，并在 '证明' 类题目中避免失分。
1. Basic Vocabulary of a Circle | 圆的基本术语
A circle is the set of points at a fixed distance from a centre. The key components are radius, chord, arc, tangent, secant, sector and segment. A central angle has its vertex at the centre, while an inscribed angle or circumference angle has its vertex on the circle. Knowing these terms helps you read diagrams accurately.
圆是与圆心距离固定的点的集合。关键元素包括半径、弦、弧、切线、割线、扇形和弓形。圆心角顶点在圆心，圆周角顶点在圆上。掌握这些术语有助于准确读图。
- Radius: OA — distance from centre to circumference. | 半径：OA——圆心到圆周的距离。
- Chord: AB — a line segment joining two points on the circle. | 弦：AB——连接圆上两点的线段。
- Tangent: a line touching the circle at exactly one point. | 切线：与圆恰好只有一个交点的直线。
- Arc: a part of the circumference. | 弧：圆周的一部分。
- Central angle: ∠AOB with vertex at O. | 圆心角：顶点在 O 的 ∠AOB。
- Inscribed angle: ∠APB with vertex P on the circle. | 圆周角：顶点 P 在圆上的 ∠APB。
In IGCSE questions, you often need to identify whether an angle is at the centre or on the circumference before applying a theorem.
在 IGCSE 题目中，你通常需要先判断一个角是圆心角还是圆周角，然后再应用定理。
2. The Centre Angle and the Circumference Angle | 圆心角与圆周角
The first key theorem states that the angle at the centre is twice the angle at the circumference when both angles are subtended by the same arc. This is the most important tool for finding missing angles in a circle.
第一个核心定理是：当圆心角和圆周角都由同一条弧所对时，圆心角等于圆周角的两倍。这是求圆中未知角最重要的工具。
∠AOB = 2 × ∠APB (same arc AB)
In the diagram, if the minor arc AB gives a circumference angle of 35°, the central angle must be 70°. This relationship works with both minor and major arcs, but you must use the same arc for both angles.
在图中，若弧 AB 所对的圆周角为 35°，则圆心角一定是 70°。这个关系对劣弧和优弧都成立，但两个角必须由同一条弧所对。
3. Angles in the Same Segment | 同弓形上的圆周角
Angles in the same segment are equal. This means that if two inscribed angles are subtended by the same chord and lie on the same side of the chord, they have the same measure. The theorem is often used when a diagram has several points on the circumference.
同弓形中的圆周角相等。也就是说，如果两个圆周角由同一条弦所对，并且位于弦的同一侧，那么它们的度数相同。当图中圆上有多个点时，常常会用到这个定理。
∠APB = ∠AQB (angles in the same segment AB)
Careful: if the two points lie on opposite sides of the chord AB, the angles are supplementary only if the points make a cyclic quadrilateral, not automatically equal. In the same segment version, both points P and Q are on the same arc AB.
注意：如果两个点位于弦 AB 的两侧，那么这些角只有在构成圆内接四边形时才会互补，而不是自动相等。在同弓形定理中，点 P 和 Q 都位于同一条弧 AB 上。
4. Angle in a Semicircle | 半圆中的角
The angle in a semicircle is always a right angle. This is a special case of the centre-angle theorem: a diameter subtends a central angle of 180°, so any circumference angle subtended by the diameter must be half of 180°, namely 90°.
半圆中的角始终是直角。这是圆心角定理的特例：直径所对的圆心角为 180°，因此直径所对的任意圆周角一定是 180° 的一半，即 90°。
∠ACB = 90° (AB is a diameter)
This theorem is extremely useful for proving that a triangle is right-angled or for finding a missing side with Pythagoras. In exam diagrams, look for a diameter marked through the centre O.
这个定理在证明三角形是直角三角形或用勾股定理求未知边时非常有用。在考试图中，要注意寻找通过圆心 O 标出的直径。
5. Cyclic Quadrilater
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