📚 Angle Properties of Circles | 圆的角度性质
In the IGCSE Extended Mathematics syllabus, the circle is far more than a simple closed curve - it is a rich source of angle relationships that examiners love to test. Circle theorems give you a powerful toolkit for solving multi-step "angle chasing" problems, from short multiple-choice questions to longer proof-style items. This article explains every key theorem, shows you how to apply it step by step, and warns you against the most common traps.
在 IGCSE 扩展数学考纲中，圆远不止是一条简单的闭合曲线，它蕴含丰富的角度关系，是命题者非常青睐的考点。圆的相关定理为你提供了强大的工具，可用于解决多步的角度推理问题，无论是简短的选择题还是较长的证明题。本文系统讲解每一条核心定理，一步一步示范应用方法，并帮助你避开最常见的失分陷阱。
1. Basic Terminology of Circles | 圆的基本术语
Before applying any theorem, you must be confident with the vocabulary. A chord is a straight line joining two points on the circumference; an arc is part of the circumference; a segment is the region bounded by a chord and an arc; a tangent touches the circle at exactly one point; a cyclic quadrilateral has all four vertices on the circle.
在运用任何定理之前，你必须熟练掌握相关术语。弦（chord）是连接圆周上两点的线段；弧（arc）是圆周的一部分；弓形（segment）是由弦和弧围成的区域；切线（tangent）与圆只有一个公共点；圆内接四边形（cyclic quadrilateral）的四个顶点都在圆上。
- Radius: a line from the centre to the circumference. 半径：从圆心到圆周的线段。
- Diameter: a chord that passes through the centre; it is twice the radius. 直径：经过圆心的弦，长度是半径的两倍。
- Chord: a straight line joining two points on the circle. 弦：连接圆周上两点的线段。
- Arc: a part of the circumference, named as arc AB. 弧：圆周的一部分，记作弧 AB。
- Tangent: a straight line that touches the circle at one point only. 切线：与圆只有一个公共点的直线。
- Cyclic quadrilateral: a quadrilateral whose four vertices lie on the circle. 圆内接四边形：四个顶点都在同一个圆上的四边形。
2. Angle in a Semicircle | 半圆内的角
The most famous theorem states that if a triangle is constructed on a diameter, the angle opposite the diameter is always a right angle. If AB is a diameter of a circle and P is any other point on the circumference, then angle APB equals 90 degrees.
其中最著名的定理是：如果以直径为边构造三角形，则直径所对的角恒为直角。设 AB 为圆的直径，P 为圆周上任意一点，则 ∠APB 等于 90°。
∠APB = 90°
Why is this true? The central angle ∠ rc AB, and any inscribed angle is half of the central angle subtending the same arc. Hence ∠APB = ½ × 180° = 90°.
为什么成立？因为 AOB 是直线，所以圆心角 ∠AOB 为 180°。点 P 处的圆周角对应同一段弧 AB，而圆周角恒等于同弧所对圆心角的一半，因此 ∠APB = ½ × 180° = 90°。
3. Angles Subtended by the Same Arc | 同弧所对的圆周角相等
on the same side of the chord. For example, if points P and Q lie on the same arc AB, then ∠APB = ∠AQB.
如果两个圆周角对应同一条弦或同一段弧，且顶点在弦的同一侧，那么这两个角相等。例如，若点 P 和 Q 在弧 AB 同侧，则 ∠APB = ∠AQB。
∠APB = ∠AQB
Closely related is the central angle theorem: the angle at the centre is twice the inscribed angle subtending the same arc. If O is the centre, then ∠AOB = 2∠APB.
与之密切相关的是圆心角定理：同弧所对的圆心角等于圆周角的两倍。若 O 为圆心，则 ∠AOB = 2∠APB。
∠AOB = 2∠APB
This theorem is the engine behind many IGCSE questions. Whenever you see an angle at the centre and an angle at the circumference sharing the same endpoints, immediately think "double or half".
这条定理是许多 IGCSE 题目的核心。每当你看到圆心角和圆周角具有相同的两个端点时，立刻联想到"两倍"或"一半"的关系。
4. Cyclic Quadrilaterals | 圆内接四边形
A cyclic quadrilateral is a quadrilateral drawn inside a circle with all four vertices on the circumference. Its most important property is that opposite angles sum to 180 degrees.
圆内接四边形是四个顶点都在圆周上的四边形。它最重要的性质是：对角之和等于 180°。
∠a + ∠c = 180°, ∠b + ∠d = 180°
In addition, an exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. These two rules are frequently used together in "find the unknown angle" questions.
此外，圆内接四边形的一个外角等于其内对角。这两个规则经常结合用于求解未知角的问题。
- Opposite angles sum to 180°. 对角之和为 180°。
- Exterior angle equals the interior opposite angle. 外角等于内对角。
5. Tangent–Radius Theorem | 切线与半径定理
A tangent to a circle is perpendicular to the radius drawn to the point of contact. If the tangent touches the circle at T and O is the centre, then the radius OT is perpendicular to the tangent line.
圆的切线垂直于过切点的半径。若切线在点 T 与圆相切，O 为圆心，则半径 OT 垂直于切线。
OT ⊥ tangent at T
= PB. This equality can help establish isosceles triangles and symmetry in many problems.
该定理还引出一个常用结论：从圆外一点 P 引两条切线，切点分别为 A 和 B，则 PA = PB。这一等量关系可用于构造等腰三角形和对称性，在解题中非常有用。
6. Alternate Segment Theorem | 弦切角定理
The alternate segment theorem is one of the most elegant results in circle geometry. It states that the angle between a tangent and a chord drawn from the point of contact is equal to the angle subtended by that chord in the opposite segment of the circle.
弦切角定理是圆几何中最优美的结论之一。它指出：切线与过切点的弦所夹的角，等于该弦所对的、位于圆另一侧的圆周角。
∠ between tangent and chord = ∠ in the alternate segment
For example, if the tangent at point P meets the chord PR, then the angle between them equals the angle subtended by PR at any point on the opposite arc of the circle. When you see a tangent and a chord sharing the same point, always check whether the alternate segment theorem applies.
例如，若点 P 处的切线与弦 PR 相交，则它们之间的夹角等于弦 PR 在圆另一侧任意一点处所张的角。当你看到切线与弦共点时，务必检查是否适用弦切角定理。
7. Perpendicular from Centre to Chord | 圆心到弦的垂线
If a radius is drawn perpendicular to a chord, it bisects the chord. Conversely, the perpendicular bisector of any chord passes through the centre of the circle.
如果半径垂直于弦，则半径平分该弦。反过来，任何弦的垂直平分线都经过圆心。
If OM ⊥ chord AB, then AM = MB
This theorem is especially useful when combined with Pythagoras' theorem. If you know the radius of the circle and the distance from the centre to a chord, you can find the length of the chord by constructing a right-angled triangle.
这条定理在结合勾股定理时特别有用。若已知圆的半径以及圆心到弦的距离，则可以通过构造直角三角形求出弦的长度。
- Perpendicular from centre bisects the chord. 圆心对弦作垂线，垂足平分弦。
- Use with Pythagoras: OM² + AM² = OA². 结合勾股定理：OM² + AM² = OA²。
8. Intersecting Chords Theorem | 相交弦定理
When two chords AB and CD intersect at a point P inside a circle, the products of the two segments of each chord are equal. This gives a very clean algebraic equation that can be solved for an unknown length.
当圆内有两条弦 AB 和 CD 相交于点 P 时，每条弦被分成的两段的乘积相等。由此得到一个简洁的代数方程，可用来求解未知长度。
AP × PB = CP × PD
Note that P is the intersection point inside the circle, and each side of the equation multiplies the two pieces of one chord. Remember to label the segments carefully before substituting values.
注意 P 是圆内的交点，等式两边分别是一条弦被分成的两段之积。代入数值前务必小心标注各段长度。
9. Worked Examples | 典型例题精讲
The best way to master circle theorems is to see them in action. Below are three examination-style problems solved step by step.
掌握圆定理最好的方法就是通过例题来体会。下面三道考试风格的题目均有逐步解答。
Example 1 | 例1：A, B and C are points on a circle with centre O. The central angle ∠AOB = 70°. Find ∠ACB, where C is on the major arc AB.
例1：A、B、C 是圆心为 O 的圆上的点，圆心角 ∠AOB = 70°，C 在优弧 AB 上，求 ∠ACB。
Solution: ∠ACB is an inscribed angle subtending the same arc AB as the central angle ∠AOB. Using the central angle theorem, ∠ACB = ½ × 70° = 35°.
解：∠ACB 与圆心角 ∠AOB 对应同一段弧 AB。由圆心角定理，∠ACB = ½ × 70° = 35°。
∠ACB = 35°
Example 2 | 例2：PQRS is a cyclic quadrilateral with ∠P = (2x + 10)° and ∠R = (3x − 20)°. Find the value of x.
例2：PQRS 是圆内接四边形，∠P = (2x + 10)°，∠R = (3x − 20)°，求 x 的值。
(2x + 10) + (3x − 20) = 180. This simplifies to 5x − 10 = 180, giving 5x = 190, hence x =
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