IGCSE Maths: Circle Theorems | IGCSE 数学：圆定理

📚 IGCSE Maths: Circle Theorems | IGCSE 数学：圆定理

Circle theorems are a central part of the IGCSE Mathematics syllabus. They describe the relationships between angles, chords, tangents and radii in a circle. Mastering these rules allows you to find missing angles quickly and to write clear geometric proofs.

圆定理是 IGCSE 数学大纲的核心内容。它们描述了圆中角、弦、切线和半径之间的关系。掌握这些规则可以让你快速求出未知角，并写出清晰的几何证明。

1. Angle at the Centre | 圆心角

For any arc AB, the angle that the arc subtends at the centre O is twice the angle it subtends at any point C on the circumference. This is often written as ∠AOB = 2 × ∠ACB, where C and O are on the same side of chord AB.

对于任意弧 AB，弧在圆心 O 处所对的圆心角是它在圆周上任意点 C 处所对圆周角的两倍。通常写作 ∠AOB = 2 × ∠ACB，其中 C 与 O 在弦 AB 的同侧。

∠AOB = 2 × ∠ACB

This theorem is the foundation for several other results. If you know the obtuse central angle, be careful: the corresponding circumference angle on the major arc is half of the reflex angle at the centre. For example, if the reflex ∠AOB = 240°, then ∠ACB = 120° on the major arc.

这个定理是其他几个结论的基础。如果你知道的是钝角圆心角，要注意：在优弧上的对应圆周角等于圆心处优角（大于 180° 的角）的一半。例如，若优角 ∠AOB = 240°，则优弧上的 ∠ACB = 120°。

2. Angle in a Semicircle | 半圆上的圆周角

When AB is a diameter of the circle, the central angle ∠AOB is exactly 180°. By the angle at the centre theorem, any angle ∠ACB subtended by the diameter at the circumference is 180° ÷ 2 = 90°.

当 AB 是圆的直径时，圆心角 ∠AOB 正好是 180°。根据圆心角定理，直径在圆周上任意点 C 处所对的圆周角 ∠ACB 都等于 180° ÷ 2 = 90°。

AB is a diameter ⇒ ∠ACB = 90°

This is a quick way to identify a right angle inside a circle. It is often used in problems where a triangle is inscribed with one side as the diameter. If a question shows a circle with a diameter and a point on the circumference, you can immediately mark a right angle.

这是在圆内识别直角的快速方法。当圆内接三角形的一条边是直径时，该定理经常被用到。如果题目中给出了圆的直径和圆周上的一点，你可以立即标出直角。

3. Angles in the Same Segment | 同弧上的圆周角

Angles subtended by the same chord or by the same arc, and lying in the same segment, are equal. In the diagram, points C and D are on the same side of chord AB, so ∠ACB = ∠ADB.

由同一条弦或同一段弧所对、且位于同一弓形内的圆周角相等。在图中，点 C 和 D 在弦 AB 的同侧，所以 ∠ACB = ∠ADB。

∠ACB = ∠ADB

Do not confuse this with angles in different segments. For a chord AB, an angle in the major segment and an angle in the minor segment are not necessarily equal. They are supplementary only when the four points form a cyclic quadrilateral, which is covered next.

不要把这个定理与不同弓形内的角混淆。对于弦 AB，优弓形内的角和劣弓形内的角不一定相等。只有当四个点构成圆内接四边形时，它们才互补，这一点将在下一节讨论。

4. Cyclic Quadrilaterals | 圆内接四边形

A cyclic quadrilateral is a four-sided figure whose vertices all lie on the circumference of a circle. The key rule is that opposite angles add up to 180°: ∠A + ∠C = 180° and ∠B + ∠D = 180°.

圆内接四边形是指四个顶点都在同一个圆上的四边形。关键规则是对角互补：∠A + ∠C = 180°，∠B + ∠D = 180°。

∠A + ∠C = 180° and ∠B + ∠D = 180°

This theorem also works in reverse: if a quadrilateral has a pair of opposite angles that sum to 180°, then its vertices are concyclic. This converse is useful in proving that four points lie on a circle.

这个定理反过来也成立：如果一个四边形有一组对角之和为 180°，那么它的四个顶点共圆。这个逆定理在证明四点共圆时非常有用。

For example, if three angles of a cyclic quadrilateral are 70°, 80° and 110°, then the fourth angle must be 100° because opposite pairs sum to 180°.

例如，如果一个圆内接四边形的三个角分别为 70°、80° 和 110°，那么第四个角一定是 100°，因为

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