Circle Theorems for IGCSE Mathematics | IGCSE 数学中的圆定理

📚 Circle Theorems for IGCSE Mathematics | IGCSE 数学中的圆定理

Circle theorems are a vital part of IGCSE Mathematics. They describe the relationships between angles, chords, radii, tangents and arcs inside a circle. Mastering these rules allows you to solve many geometry questions quickly and confidently.

圆定理是 IGCSE 数学的重要组成部分。它们描述了圆中角、弦、半径、切线与弧之间的关系。熟练掌握这些规则，能帮助你快速而自信地解决许多几何题目。

1. Key Parts of a Circle | 圆的基本元素

Before applying circle theorems, you must be familiar with the key components. The centre is the fixed point inside the circle. A radius is a line segment from the centre to any point on the circumference.

在应用圆定理之前，你必须熟悉圆的基本组成。圆心是圆内的固定点；半径是从圆心到圆周上任意一点的线段。

A diameter is a chord that passes through the centre and is twice the radius. A chord is any straight line joining two points on the circumference. An arc is part of the circumference, and a tangent is a line that touches the circle at exactly one point.

直径是经过圆心且长度为半径两倍的弦；弦是连接圆周上两点的任意直线段；弧是圆周的一部分；切线是与圆恰好只有一个交点的直线。

Radius = half of the diameter.
All radii in the same circle are equal.
A diameter is the longest chord of a circle.

半径等于直径的一半。
同一个圆中的所有半径都相等。
直径是圆中最长的弦。

2. Angle in a Semicircle | 半圆上的圆周角

If a triangle is drawn inside a circle such that one of its sides is the diameter, then the angle opposite the diameter is a right angle. This is called the angle in a semicircle theorem.

如果在一个圆内画一个三角形，且三角形的一条边是直径，那么直径所对的角是直角。这被称为“半圆上的圆周角定理”。

If AOB is a diameter, then ∠ACB = 90°

Here point C is any point on the circumference. This theorem is especially useful when you need to identify a right angle before using Pythagoras’ theorem or trigonometry.

这里点 C 是圆周上的任意一点。该定理特别有用，因为当你需要先找出一个直角，再使用勾股定理或三角函数时，可以直接依托直径。

3. The Central Angle Theorem | 圆心角定理

The central angle theorem states that the angle at the centre of a circle is twice the angle at the circumference, when both angles stand on the same arc.

圆心角定理指出：当圆心角和圆周角对应同一段弧时，圆心角等于圆周角的两倍。

∠AOB = 2 × ∠ACB

In this diagram, points A and B are on the circumference, and point C is also on the circumference. Angle AOB is formed by two radii, while angle ACB is formed by two chords.

在图中，点 A 和 B 在圆周上，点 C 也在圆周上。∠AOB 由两条半径构成，而∠ACB 由两条弦构成。

You must make sure that both angles stand on the same arc AB. If C lies on the major arc, use the minor arc AB; if C lies on the minor arc, use the major arc AB.

你需要注意，两个角必须对应同一段弧 AB。如果点 C 位于优弧上，就对应劣弧 AB；如果点 C 位于劣弧上，就对应优弧 AB。

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

Angles subtended by the same chord at the circumference are equal. This is known as the “same segment” theorem.

同一弦在圆周上所形成的所有圆周角都相等。这被称为“同弧上的圆周角相等”定理。

If ∠ACB and ∠ADB stand on chord AB, then ∠ACB = ∠ADB

Points C and D must lie on the same side of chord AB. If one point is on the opposite side, the angles will sum to 180° instead.

点 C 和点 D 必须位于弦 AB 的同侧。如果一点在另一侧，那么这两个角之和将等于 180°。

This theorem is frequently used in IGCSE exams with other rules, such as the isosceles triangle formed by two radii.

在 IGCSE 考试中，这个定理经常与其他规则一起使用，例如由两条半径构成的等腰三角形。

5. Cyclic Quadrilaterals | 圆内接四边形

A cyclic quadrilateral is a four-sided shape with all four vertices on the circumference of a circle. One key property is that opposite angles sum to 180°.

圆内接四边形是指四个顶点都在同一个圆上的四边形。它的一个重要性质是：对角之和等于 180°。

∠a + ∠c = 180° , ∠b + ∠d = 180°

Another useful fact is that the exterior angle of a cyclic quadrilateral is equal to the opposite interior angle. This can simplify many angle chasing questions.

另一个有用的事实是：圆内接四边形的外角等于其不相邻的内角。这个性质可以简化许多角度计算题。

When you see a quadrilateral drawn inside a circle, always check whether it is cyclic. If it is, write down the opposite angle sums before solving.

当你看到一个四边形画在圆内时，先判断它是否为圆内接四边形。如果是，先写下对角之和等于 180° 的关系，再继续解题。

6. Tangent and Radius | 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 fundamental circle theorem.

圆的切线垂直于经过切点的半径。这是一个基础的圆定理。

If OT is a radius and T is the point of contact, then OT ⊥ tangent at T

Therefore, the angle between a radius and a tangent is always 90°. This right angle is often the key to solving problems that involve tangents.

因此，半径与切线之间的角总是 90°。这个直角通常是解决涉及切线的题目的关键。

Another important property: if two tangents are drawn to a circle from an external point, the tangent segments are equal in length.

另一个重要性质：从圆外一点引圆的两条切线，则两条切线段长度相等。

PA = PB , where P is the external point

This equal-length property creates two congruent right-angled triangles, which can be used to find unknown lengths and angles.

这个等长性质会形成两个全等的直角三角形，可用于求未知的长度和角度。

7. Alternate Segment Theorem | 弦切角定理

The alternate segment theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment of the circle.

弦切角定理指出：切线与经过切点的弦所成的角，等于该弦所对的圆内“交替弧段”上的圆周角。

∠BTD = ∠BAT , where TD is a tangent at T

In other words, the angle made by the tangent and chord at point T equals any angle in the opposite arc of chord BT.

换句话说，在点 T 处切线与弦 BT 所成的角，等于弦 BT 所对的另一侧弧上的任意圆周角。

To apply this theorem correctly, first identify the tangent and the chord. Then look for the angle in the segment on the opposite side of that chord.

要正确运用这个定理，首先要找出切线和弦。然后寻找该弦另一侧弧段上的圆周角。

8. Equal Chords and Arcs | 等弦与等弧

Equal chords in a circle subtend equal angles at the centre. They also subtend equal angles at the circumference.

在同圆或等圆中，等弦所对的圆心角相等，并且所对的圆周角也相等。

If chord AB = chord CD, then ∠AOB = ∠COD

Also, equal chords are at equal perpendicular distances from the centre of the circle.

此外，等弦到圆心的垂直距离也相等。

Equal chords cut off equal arcs.
The perpendicular from the centre to a chord bisects the chord.

等弦所截的弧相等。
圆心到弦的垂线平分这条弦。

The perpendicular bisector of any chord always passes through the centre of the circle. This fact helps locate the centre in construction problems.

任意弦的垂直平分线一定经过圆心。这一事实可以帮助你在作图问题中确定圆心。

9. Problem-Solving with Circle Theorems | 用圆定理解题

When faced with a circle theorem question, first mark any known angles and equal lengths. Look for a right angle created by a diameter or by a tangent and radius.

遇到圆定理题目时，先标出已知角度和相等长度。寻找直径或切线与半径所产生的直角。

Then identify which theorem connects the given information to the unknown angle. For example, if you see an angle at the centre and an angle at the circumference, use the central angle theorem.

然后判断哪个定理能将已知信息与未知角联系起来。例如，如果同时看到一个圆心角和一个圆周角，就使用圆心角定理。

Worked example: In a circle, ∠AOB = 70° and C is a point on the circumference. Find ∠ACB.

示例：在一个圆中，∠AOB = 70°，点 C 在圆周上。求∠ACB。

Since the angle at the centre is twice the angle at the circumference on the same arc:

因为同一弧上的圆心角等于圆周角的两倍：

∠ACB = 70° ÷ 2 = 35°

Always write the reason you used each step in your working. In the exam, a correct reason earns a method mark.

解题过程中每一步都要写出理由。在考试中，正确的理由可以获得方法分。

10. Summary and Exam Tips | 总结与考试技巧

Here is a quick review of the circle theorems you must remember for IGCSE Mathematics:

以下是你必须为 IGCSE 数学记住的圆定理快速回顾：

A diameter subtends a right angle.
Angle at centre = 2 × angle at circumference.
Angles in the same segment are equal.
Opposite angles of a cyclic quadrilateral sum to 180°.
Radius and tangent are perpendicular.
Tangents from an external point are equal.
Alternate segment theorem: tangent-chord angle equals angle in the alternate segment.

直径所对的圆周角为直角。
圆心角等于圆周角的两倍。
同弧上的圆周角相等。
圆内接四边形的对角互补，和为 180°。
半径与切线互相垂直。
从圆外一点引出的两条切线长相等。
弦切角定理：切线与弦的夹角等于交替弧段上的圆周角。

In the exam, draw a clear diagram and label every angle you find. Do not assume an angle is a right angle unless you can justify it with a theorem or given data.

考试中，请画清晰的图，并标注每一个求出的角。除非你能用定理或已知条件证明一个角是直角，否则不要擅自假设。

Practice past-paper questions until you can recognise each theorem quickly. The more familiar you are, the faster you will solve these problems.

多练习历年真题，直到你能快速识别每个定理。你越熟悉

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Circle Theorems for IGCSE Mathematics | IGCSE 数学中的圆定理