Circle Theorems | 圆定理

📚 Circle Theorems | 圆定理

Welcome to this revision article on circle theorems, a key topic in IGCSE Mathematics. These powerful rules connect angles, chords, tangents and arcs inside a circle. Once you understand them, many seemingly complex geometry problems become straightforward.

欢迎阅读这篇关于圆定理的复习文章，圆定理是IGCSE数学中的一个重要考点。这些强有力的法则将圆内的角、弦、切线和弧联系起来。一旦理解了它们，许多看似复杂的几何问题就会变得简单直接。

1. Basic Terms of a Circle | 圆的基本术语

Before using the theorems, you must be familiar with the following parts of a circle: centre, radius, diameter, chord, arc, tangent, inscribed angle and central angle.

在使用这些定理之前，你必须熟悉圆的以下组成部分：圆心、半径、直径、弦、弧、切线、圆周角和圆心角。

• Radius: a line from the centre to any point on the circle.

  半径：从圆心到圆上任意一点的线段。

• Chord: a line segment joining two points on the circle; the diameter is the longest chord.

  弦：连接圆上两点的线段；直径是最长的弦。

• Arc: part of the circumference between two points.

  弧：圆周上两点之间的部分。

• 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。

These terms will be used constantly in the theorems below.

下面的定理将反复使用这些术语。

2. The Central Angle is Twice the Inscribed Angle | 圆心角是圆周角的两倍

For any arc AB, the central angle ∠AOB is twice the inscribed angle ∠ACB that subtends the same arc.

对于任意一条弧AB，圆心角∠AOB等于同一条弧所对圆周角∠ACB的两倍。

∠AOB = 2 × ∠ACB

This is the most fundamental circle theorem and many other theorems follow from it.

这是最基本的圆定理，许多其他定理都由它推导而来。

3. Angle in a Semicircle is 90° | 半圆所对的圆周角是90°

If AB is a diameter of the circle, then the angle at any point C on the circumference is a right angle.

如果AB是圆的直径，那么圆上任意一点C处所对的圆周角∠ACB都是直角。

If AB is a diameter, then ∠ACB = 90°

This case is simply a special application of Theorem 2, since the central angle ∠AOB is 180°.

这个情形是定理2的特殊应用，因为此时圆心角∠AOB为180°。

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

Angles subtended by the same chord (or the same arc) at the circumference are equal.

同一条弦（或同一条弧）所对的圆周角相等。

If ∠ACB and ∠ADB are both subtended by chord AB, then ∠ACB = ∠ADB

This means that when two points C and D lie on the same side of chord AB, the angles they form with A and B are equal.

这意味着当点C和点D在弦AB的同侧时，它们与A、B构成的角相等。

5. Equal Arcs Subtend Equal Angles | 等弧所对的角相等

In the same circle, equal arcs subtend equal angles at the centre and at the circumference.

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