IGCSE Mathematics Topic 219: Circle Theorems | IGCSE 数学主题 219：圆定理

📚 IGCSE Mathematics Topic 219: Circle Theorems | IGCSE 数学主题 219：圆定理

Circle theorems are a cornerstone of IGCSE geometry. They describe the relationships between angles, lines, and points on a circle. In this revision guide we break down every theorem you need for your Edexcel exam, explain why it works, and show you how to apply it confidently.

圆定理是 IGCSE 几何的核心内容。它们描述了圆上角度、直线和点之间的关系。在本复习指南中，我们将逐一拆解你在 Edexcel 考试中需要掌握的所有定理，解释其原理，并展示如何自信地应用它们。

1. Circles: Key Terms and Symbols | 圆：关键术语与符号

Before we dive into the theorems, make sure you know the essential vocabulary. A circle is a set of points equidistant from a fixed point called the centre. The radius (plural radii) is a line segment from the centre to any point on the circle. A chord is a line segment joining two points on the circumference. The diameter is a chord that passes through the centre and is twice the length of the radius.

在深入定理之前，请确保你掌握关键术语。圆是到圆心这个固定点距离相等的点的集合。半径（复数 radii）是从圆心到圆上任意一点的线段。弦是连接圆周上两点的线段。直径是经过圆心的弦，长度是半径的两倍。

• Arc: part of the circumference between two points. | 弧：两点之间圆周的一部分。

• Sector: a region bounded by two radii and an arc. | 扇形：由两条半径和一段弧围成的区域。

• Segment: a region bounded by a chord and the arc it cuts off. | 弓形：由一条弦和它所截的弧围成的区域。

• Tangent: a straight line that touches the circle at exactly one point. | 切线：与圆恰好只有一个交点的直线。

2. Angle at the Centre is Twice the Angle at the Circumference | 圆心角是圆周角的两倍

The most important circle theorem states that the angle subtended by an arc at the centre of the circle is twice the angle subtended by the same arc at any point on the circumference. In the diagram, if points A, B and P lie on the circle with O as the centre, then ∠AOB = 2 × ∠APB.

最重要的圆定理是：同一条弧所对的圆心角等于它所对的圆周角的两倍。在图中，如果 A、B、P 三点在圆上，O 为圆心，则 ∠AOB = 2 × ∠APB。

∠AOB = 2 ∠APB

This theorem works for every position of P on the circumference, as long as P is on the same arc AB. If P is on the major arc, the smaller angle at the centre is used; if P is on the minor arc, the reflex angle at the centre is used.

只要 P 在弧 AB 上，无论 P 在圆周上的哪个位置，该定理都成立。如果 P 在优弧上，取较小的圆心角；如果 P 在劣弧上，则取较大的反射角。

3. Angle in a Semicircle is a Right Angle | 半圆中的角是直角

A special case of the previous theorem happens when the arc is a semicircle. If AB is the diameter of a circle and P is any point on the circumference, then the angle ∠APB is always 90°.

前一定理的一个特殊情况是当弧为半圆时。如果 AB 是圆的直径，P 是圆周上任意一点，则 ∠APB 恒等于 90°。

If AB is a diameter, then ∠APB = 90°

This theorem is extremely useful in coordinate geometry and constructions. When you see a diameter, look for a right angle immediately.

该定理在坐标几何和作图中非常有用。当你看到直径时，应立刻寻找直角。

4. Angles in the Same Segment are Equal | 同弧上的圆周角相等

If two angles are subtended by the same chord, and their vertices lie on the same side of the chord, the angles are equal. In other words, points A, B, P and Q are on the circle, and P and Q are on the same arc AB, then ∠APB = ∠AQB.

如果两个圆周角对应同一条弦，且顶点在弦的同一侧，那么这两个角相等。换句话说，A、B、P、Q 在圆上，且 P、Q 在弧 AB 上，则 ∠APB = ∠AQB。

∠APB = ∠AQB

When solving problems, look for multiple triangles sharing the same base. Mark all equal angles to avoid missing a hidden relationship.

解题时，寻找共用底边的多个三角形。标记所有相等的角，以免遗漏隐藏的关系。

5. Opposite Angles in a Cyclic Quadrilateral Sum to 180° | 圆内接四边形对角互补

A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the same circle. In any cyclic quadrilateral, the sum of each pair of opposite angles is 180°.

圆内接四边形是四个顶点都在同一个圆上的四边形。在任意圆内接四边形中，每一对对角之和等于 180°。

∠A + ∠C = 180°, ∠B + ∠D = 180°

This theorem is often tested together with angle chasing in a circle. Remember that an exterior angle of a cyclic quadrilateral is equal to the interior opposite angle, which is also useful in proofs.

该定理常与圆中的角度推理一起考查。记住圆内接四边形的一个外角等于其内对角，这在证明中也很有用。

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