📚 Mastering Circle Theorems for IGCSE Mathematics | IGCSE 数学圆定理精通
Circle theorems are a cornerstone of IGCSE geometry. They describe angle and length relationships in circles and are frequently tested in both structured questions and multi-step proofs. A solid understanding of these rules will help you solve problems quickly and accurately.
圆定理是 IGCSE 几何的核心内容。它们描述了圆中角与边的关系，经常出现在结构化问题和多步证明中。扎实掌握这些规则有助于你快速、准确地解题。
1. Angle at the Centre Theorem | 圆心角定理
If an arc AB subtends an angle at the centre O and an angle at the circumference P, then the central angle is twice the circumference angle.
如果弧 AB 在圆心 O 处所对的角和在圆周 P 处所对的角，那么圆心角是圆周角的两倍。
∠AOB = 2 × ∠APB
This theorem applies only when both angles stand on the same arc. For example, if ∠APB = 35°, then ∠AOB = 70°.
该定理仅适用于两个角对着同一条弧的情况。例如，如果 ∠APB = 35°，那么 ∠AOB = 70°。
and the circumference angle as x + 20, you can write 3x + 10 = 2(x + 20) and solve for x.
在许多题目中，你需要利用这个关系建立方程。如果圆心角给定为 3x + 10，圆周角给定为 x + 20，你可以写出 3x + 10 = 2(x + 20) 并解出 x。
2. Angle in a Semicircle | 半圆中的圆周角
Any angle subtended by a diameter at the circumference is a right angle (90°).
直径在圆周上所对的任意角都是直角（90°）。
∠APB = 90° when AB is a diameter
This is a special case of the angle at the centre theorem: a diameter creates a central angle of 180°, so the angle at the circumference is half of that, 90°.
这是圆心角定理的特殊情况：直径产生的圆心角为 180°，因此圆周角是它的一半，即 90°。
The converse is also useful: if an angle subtended by a chord at the circumference is 90°, then the chord must be a diameter of the circle.
其逆命题也很有用：如果一条弦在圆周上所对的角是 90°，那么这条弦一定是圆的直径。
3. Angles in the Same Segment | 同弓形中的圆周角
Angles subtended by the same chord or arc and lying in the same segment are equal.
同一条弦或弧所对的、位于同一弓形内的圆周角相等。
∠APB = ∠AQB
In diagrams with several triangles sharing a base, this rule lets you transfer an angle from one point to another on the same side of the chord. It is especially common in cyclic quadrilateral and circle geometry problems.
在多个三角形共用一条底边的图形中，此规则可让你将角从弦同侧的一点转移到另一点。它在圆内接四边形和圆几何问题中尤其常见。
For example, if P and Q are two points on the major arc AB, then ∠APB and ∠AQB are equal, even if the triangles look different.
例如，如果 P 和 Q 是优弧 AB 上的两个点，那么 ∠APB 和 ∠AQB 相等，即使三角形看起来不同。
4. Cyclic
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