📚 IGCSE Mathematics: Circle Theorems Made Simple | IGCSE 数学：轻松掌握圆定理
Circle theorems appear regularly in IGCSE Mathematics exams, especially in Paper 2 and Paper 4 multi-step problems. They connect angles, chords, tangents and cyclic quadrilaterals. The key to success is not memorising the diagrams in isolation: you need to recognise which theorem fits the given diagram and then set out a clear angle chase. This guide walks through the main circle theorems, each with a statement, a simple proof or reason, an example, and common mistakes to avoid.
圆定理是 IGCSE 数学考试中的常见考点，尤其是在 Paper 2 和 Paper 4 的多步骤综合题中。它们把角、弦、切线和圆内接四边形联系在一起。学好圆定理的关键不是孤立地记住图形，而是能够根据题目图形识别适用的定理，并清晰地写出角度推导过程。本指南逐一讲解主要圆定理，每个定理都包含内容、简单理由或证明、例题以及常见错误提醒。
1. The Angle at the Centre Is Twice the Angle at the Circumference | 圆心角是圆周角的两倍
ended at the centre by arc AB is twice the angle subtended at the circumference by the same arc. In symbols, ∠AOB = 2 × ∠ACB, where C lies on the remaining part of the circumference.
第一个定理把圆心角与圆周角联系起来。如果 A、B、C 是圆上的点，O 是圆心，那么弧 AB 所对的圆心角是同一弧所对圆周角的两倍。用符号表示就是 ∠AOB = 2 × ∠ACB，其中 C 位于弧的另一侧圆周上。
∠AOB = 2 × ∠ACB
This rule works both ways. If you know the angle at the centre, divide by 2 to get the angle at the circumference. For example, when ∠AOB = 110°, the corresponding angle ∠ to find the central angle. Always check that both angles are standing on the same arc AB.
这个定理可以双向使用。如果已知圆心角，除以 2 就得到圆周角。例如，当 ∠AOB = 110° 时，对应的圆周角 ∠ACB = 55°。如果已知圆周角，乘以 2 就得到圆心角。做题时一定要检查两个角是否站在同一段弧 AB 上。
2. The Angle in a Semicircle Is a Right Angle | 半圆上的圆周角是直角
AB is always a right angle. This is a special case of the first theorem: the central angle ∠ O and B lie on a straight line, so the angle at the circumference is half of 180°, giving ∠APB = 90°.
如果 AB 是圆的直径，那么 AB 所对的圆周角始终是直角。这是第一个定理的特殊情况：因为 A、O、B 三点共线，所以圆心角 ∠AOB 等于 180°，圆周角是它的一半，因此 ∠APB = 90°。
∠APB = 90°
This theorem is very useful for proving that a triangle is right-angled. Once you know that a triangle inside a circle has a diameter as one side, you are allowed to use Pythagoras' theorem or basic trigonometry on the right angle. Look for a diameter marked on the diagram and an angle sitting on the circumference.
这个定理在证明三角形是直角三角形时非常有用。一旦知道圆内三角形的一条边是直径，就可以在直角处使用勾股定理或基本三角函数。解题时要留意图中是否标出了直径，以及是否有一个圆周角正好落在圆周上。
3. Angles in the Same Segment Are Equal | 同弦上的圆周角相等
Angles in the same segment are equal. This means that if P and Q are on the same side of chord AB, the angles ∠APB and ∠AQB are equal. The reason comes from the first theorem: both angles are half of the same central angle ∠AOB, so they must be equal.
同弦上的圆周角相等。意思是，如果 P 和 Q 位于弦 AB 的同一侧，那么 ∠APB 与 ∠AQB 相等。原因来自第一个定理：这两个角都是同一个圆心角 ∠AOB 的一半，所以它们一定相等。
∠APB = ∠AQB
When you see a 'bow' shape with two angles opening onto the same chord from the same side, mark them as equal. This can help you form equations or transfer one known angle to another position in a proof. Be careful: angles on opposite sides of a chord are not equal in the same way; they are supplementary if the points form a cyclic quadrilateral, which is covered in the next section.
当你看到两个角从弦的同一侧朝同一段弧张开，形成“弓形”时，可以把它们标为相等。这能帮助你列出方程，或在证明中把一个已知角转移到另一个位置。注意：弦两侧的角并不按这种方式相等；如果四个点构成圆内接四边形，它们互补，下一节会讲到。
4. Opposite Angles in a Cyclic Quadrilateral Add to 180° | 圆内接四边形对角互补
same circle. For any cyclic quadrilateral ABCD, the opposite angles add to 180°. That is, ∠A + ∠C = 180° and ∠B + ∠ comes from the fact that the opposite angles subtend complementary arcs around the circle.
圆内接四边形是指四个顶点都在同一个圆上的四边形。对于任意圆
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