IGCSE Circle Theorems: Complete Topic T-4-1028 | IGCSE 圆定理：完整专题 T-4-1028

📚 IGCSE Circle Theorems: Complete Topic T-4-1028 | IGCSE 圆定理：完整专题 T-4-1028

Circle theorems are a central part of IGCSE geometry. They describe angle and length relationships involving circles, chords, tangents, and cyclic quadrilaterals. Mastering these rules allows you to find unknown angles quickly and to justify each step in a proof-style question.

圆定理是 IGCSE 几何的核心内容。它们描述圆、弦、切线和圆内接四边形之间的角度与长度关系。掌握这些规则可以帮助你快速求出未知角，并在几何证明题中清楚地写出每一步依据。

1. Angle at the Centre / 圆心角与圆周角

The first core rule states that the angle subtended by an arc at the centre of a circle is exactly twice the angle subtended by the same arc at the circumference. This works for any arc, whether minor or major, as long as the two angles stand on the same arc.

第一条核心规则是：同一条弧所对的圆心角等于该弧所对圆周角的两倍。无论这条弧是优弧还是劣弧，只要两个角站在同一条弧上，这个关系都成立。

∠AOC = 2 × ∠ABC

In a diagram, if O is the centre and A, B, C are points on the circle, angle AOC at the centre is twice angle ABC at the circumference. You can use this to find either angle when the other is known, but be careful to identify the same arc.

在图中，如果 O 是圆心，A、B、C 是圆上的点，那么圆心角 ∠AOC 是圆周角 ∠ABC 的两倍。你可以利用这个关系由其中一个角求另一个角，但必须注意确认两个角所对的是同一条弧。

2. Angles in the Same Segment / 同弓形上的圆周角

All angles in the same segment, standing on the same chord, are equal. This means that if several angles are drawn from the same chord to different points on the same arc, their sizes are identical.

位于同一弓形内、由同一条弦所对的所有圆周角都相等。也就是说，如果一条弦对应的几个圆周角指向同一条弧上的不同点，它们的大小完全相同。

∠APB = ∠AQB

This theorem is very useful in cyclic diagrams where a chord appears to support two or more angles at the circumference. Look for the same base chord and the same side of the chord before stating equality.

这个定理在圆内接图形中非常有用，当一条弦支撑两个或更多圆周角时就会出现。使用前要先找到同一条底弦，并确认这些角位于弦的同侧。

3. Angle in a Semicircle / 半圆上的圆周角

An angle inscribed

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