Circle Theorems for IGCSE Mathematics | IGCSE 数学圆定理核心总结

📚 Circle Theorems for IGCSE Mathematics | IGCSE 数学圆定理核心总结

Circle theorems are a key part of the IGCSE Mathematics syllabus. They describe the relationships between angles, lines and lengths inside a circle. Mastering these theorems will help you solve geometry problems quickly and confidently, especially in the non-calculator paper.

圆定理是 IGCSE 数学大纲中的重点内容。它们描述了圆中角、线与长度之间的关系。掌握这些定理能帮助你快速且自信地解决几何题，尤其是在非计算器卷中。

1. Angle at the Centre Theorem | 圆心角定理

The angle at the centre of a circle is always twice any angle at the circumference subtended by the same arc. If A and B are two points on the circle and C is another point on the major arc, then the central angle ∠AOB equals 2 × ∠ACB.

圆心角总是等于同弧所对圆周角的两倍。如果 A 和 B 是圆上两点，C 是优弧上的另一点，那么圆心角 ∠AOB 等于 2 × ∠ACB。

∠AOB = 2 × ∠ACB

This theorem is often tested with diagrams where the angle at the centre and the angle at the circumference share the same chord AB. Identify the chord first, then apply the ratio 2 : 1.

这个定理经常在图中考查，其中圆心角和圆周角共用同一条弦 AB。先找到公共弦，再应用 2 : 1 的比例关系。

2. Angle in a Semicircle | 半圆上的圆周角

Any angle subtended by the diameter at the circumference is a right angle. If AB is a diameter of the circle and P lies on the circumference, then ∠APB = 90°.

直径所对的圆周角是直角。如果 AB 是圆的直径，点 P 在圆周上，那么 ∠APB = 90°。

AB is a diameter ⇒ ∠APB = 90°

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