Shape and Space 8: Circle Theorems | 图形与空间 8：圆的定理

📚 Shape and Space 8: Circle Theorems | 图形与空间 8：圆的定理

Welcome to Shape and Space 8 for the Edexcel IGCSE Mathematics course. This unit brings together circle geometry, one of the most visual and frequently tested areas of the exam. You will learn the standard circle theorems, how to apply them to find missing angles, and how to present clear logical reasons in your answers.

欢迎进入 Edexcel IGCSE 数学课程的 Shape and Space 8 单元。本单元系统讲解圆几何，这是考试中最直观也最常见的高频考点。你将学习标准圆定理，如何运用它们求未知角，以及如何在作答中写出清晰的逻辑理由。

1. Key Circle Vocabulary | 圆的基本术语

Before using the theorems, you need to recall the names of the parts of a circle. These terms appear in many exam questions and in the statements of the theorems themselves.

在运用定理之前，你需要记住圆的各个部分的名称。这些术语会出现在许多考题中，也会出现在定理本身的表述里。

• Centre (O): the point equidistant from every point on the circle.

  圆心 (O)：到圆上每一点距离都相等的点。

• Radius (r): a straight line segment from the centre to a point on the circumference.

  半径 (r)：从圆心到圆周上任意一点的线段。

• Diameter (d): a chord that passes through the centre; d = 2r.

  直径 (d)：经过圆心的弦；d = 2r。

• Chord: a line segment whose endpoints both lie on the circumference.

  弦：两端都在圆周上的线段。

• Tangent: a straight line that touches the circle at exactly one point.

  切线：与圆只有一个公共点的直线。

• Arc: part of the circumference.

  弧：圆周的一部分。

• Segment: the region bounded by a chord and an arc.

  弓形：由弦和弧围成的区域。

• Sector: the region bounded by two radii and the arc between them.

  扇形：由两条半径和它们之间的一段弧围成的区域。

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

The first circle theorem states that the angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at the circumference.

第一个圆定理指出：同一条弧所对的圆心角，等于同一条弧所对圆周角的两倍。

∠AOB = 2 × ∠ACB

In the diagram, points A and B are on the circle, O is the centre, and C is any point on the circumference. The angle ∠AOB is at the centre, while ∠ACB is at the circumference.

在图中，A 和 B 在圆上，O 是圆心，C 是圆周上任意一点。∠AOB 是圆心角，∠ACB 是圆周角。

This theorem is the basis for two important special cases. If AB is a diameter, then the angle at the centre is 180°, so the angle at the circumference is 90°.

这个定理还导出两个重要的特殊情况。如果 AB 是直径，那么圆心角为 180°，所以圆周角为 90°。

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3. Theorem 2: Angle in a Semicircle | 定理 2：半圆上的圆周角

If AB is a diameter of a circle and C is any point on the circumference, then ∠ACB = 90°.

如果 AB 是圆的直径，C 是圆周上任意一点，那么 ∠ACB = 90°。

∠ACB = 90°

This is a direct consequence of the angle at the centre theorem. Since the central angle ∠AOB is 180°, the angle at the circumference is half of it.

这是圆心角定理的直接推论。因为圆心角 ∠AOB 为 180°，所以圆周角等于它的一半。

In exam questions, always look for a diameter because it immediately tells you that there is a right angle.

在考试中，一定要留意直径，因为只要出现直径，就能立刻确定一个直角。

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4. Theorem 3: Angles in the Same Segment | 定理 3：同弧上的圆周角相等

Angles in the same segment of a circle are equal. In other words, if two points C and D lie on the same arc AB, then ∠ACB = ∠ADB.

同一弓形内的圆周角相等。也就是说，如果 C 和 D 在同一条弧 AB 上，那么 ∠ACB = ∠ADB。

∠ACB = ∠ADB

Both angles subtend the same chord AB. The chord AB divides the circle into two segments; the points C and D must be in the same segment.

这两个角都对同一条弦 AB。弦 AB 把圆分成两个弓形；点 C 和 D 必须在同一个弓形内。

This theorem is especially useful when you need to identify equal angles on a complex diagram.

这个定理在处理复杂图形时特别有用，可以帮助你快速找到相等的角。

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5. Theorem 4: Cyclic Quadrilaterals | 定理 4：圆内接四边形

A cyclic quadrilateral is a quadrilateral with all four vertices on the circumference of a circle. In a cyclic quadrilateral, the sum of each pair of opposite angles is 180°.

圆内接四边形是指四个顶点都在同一个圆上的四边形。在圆内接四边形中，任意一组对角之和等于 180°。

∠A + ∠C = 180°, ∠B + ∠D = 180°

For example, if ABCD is cyclic, then ∠A and ∠C are opposite angles, so they add up to 180°. The same is true for ∠B and ∠D.

例如，若 ABCD 为圆内接四边形，则 ∠A 与 ∠C 是对角，所以它们相加为 180°；∠B 与 ∠D 同理。

A related fact is that the exterior angle of a cyclic quadrilateral is equal to the opposite interior angle. You can use this to find angles quickly.

另一个相关结论是：圆内接四边形的一个外角等于其相对的內角。利用这一点可以快速求角。

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6. Theorem 5: Tangent and Radius | 定理 5：切线与半径垂直

A tangent to a circle is perpendicular to the radius drawn to the point of contact. This is a simple but powerful property.

圆的切线与过切点的半径互相垂直。这是一个简单但非常有力的性质。

OA ⊥ TA, so ∠OAT = 90°

If TA is a tangent at point A and O is the centre, then the radius OA is perpendicular to TA.

若 TA 是圆在 A 点处的切线，O 为圆心，则半径 OA 垂直于 TA。

This property creates a right angle, which often allows you to use Pythagoras’ theorem or angle sums in triangles.

这条性质会产生一个直角，因此常可进一步使用勾股定理或三角形内角和来求解。

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