📚 Circle Theorems | 圆的定理
Circle theorems are one of the most heavily tested geometry topics in the Edexcel IGCSE Mathematics examination. Mastering these rules allows you to solve complex angle problems quickly and confidently, and they appear regularly in both Paper 1 (non-calculator) and Paper 2 (calculator).
圆的定理（Circle Theorems）是爱德思 IGCSE 数学考试中考查频率最高的几何专题之一。熟练掌屋这些定理，可以帮助你在 Paper 1（不能用计算器）和 Paper 2（可用计算器）中快速、准确地解决各种复杂的角度计算问题。
1. Essential Vocabulary and Notation | 基础术语与符号
Before applying the theorems, you must be confident with the key words: centre (O), radius (r), diameter (d = 2r), chord, arc, segment, sector and tangent. A chord is a line segment joining two points on the circumference; an arc is part of the circumference; a tangent is a straight line that touches the circle at exactly one point.
在运用定理之前，你需要熟悉以下关键词：圆心（O）、半径（r）、直径（d = 2r）、弦、弧、弓形、扇形和切线。弦是连接圆上两点的线段；弧是圆周的一部分；切线是与圆只有一个公共点的直线。
You should also recall two basic facts: the total angle around a point is 360°, and the angle on a straight line is 180°. A radius is a line segment from the centre to any point on the circumference, and all radii of the same circle are equal in length. Whenever two radii are joined by a chord, an isosceles triangle is formed.
你还需要记住两个基本事实：一个点周围的总角度为 360°，直线上的角度为 180°。半径是从圆心到圆周上任意一点的线段，同一圆内所有半径长度相等。当两条半径与一条弦相连时，就会形成一个等腰三角形。
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Centre / 圆心: the point equidistant from every point on the circle.
圆心：与圆上每一个点距离相等的点。
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Radius / 半径: a segment from the centre to the circumference.
半径：从圆心到圆周的线段。
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Diameter / 直径: a chord that passes through the centre, d = 2r.
直径：经过圆心的弦，d = 2r。
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Chord / 弦: a segment joining two points on the circumference.
弦：连接圆周上两点的线段。
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Tangent / 切线: a straight line touching the circle at exactly one point.
切线：与圆只有一个公共点的直线。
2. Angle in a Semicircle | 半圆中的角
If A, B and C are points on a circle and AB is a diameter, then the angle at C is a right angle: ∠ACB = 90°. This is known as the angle in a semicircle theorem, and it is one of the most frequently used rules in the IGCSE examination.
如果 A、B、C 是圆上的点，且 AB 是直径，那么点 C 处的角是直角：∠ACB = 90°。这就是“半圆中的角”定理，也是 IGCSE 考试中最常用的定理之一。
The reason for this result can be seen by joining C to the centre O. Since OA = OC = OB (all are radii), triangle AOC and triangle BOC are both isosceles. Using the fact that the angles in triangle ABC add up to 180°, you can show that ∠ACB must be 90°.
这个结论可以通过连接点 C 与圆心 O 来理解。因为 OA = OC = OB（均为半径），三角形 AOC 和三角形 BOC 都是等腰三角形。利用三角形 ABC 内角和为 180° 的事实，可以证明 ∠ACB 必定等于 90°。
If AB is a diameter, then ∠ACB = 90°.
若 AB 是直径，则 ∠ACB = 90°。
3. Angle at the Centre | 圆心角定理
The angle at the centre of a circle is twice the angle at the circumference, provided both angles stand on the same arc. For example, if O is the centre and A, B, C lie on the circumference, then ∠AOB = 2 × ∠ACB.
圆周角定理指出：圆心角等于同一条弧所对圆周角的两倍。例如，如果 O 是圆心，A、B、C 在圆周上，则 ∠AOB = 2 × ∠ACB。
This theorem works for any position of C on the major or minor arc, and it is often combined with other theorems in multi-step questions. In Edexcel IGCSE papers, you will usually be asked to give a reason such as “angle at the centre is twice the angle at the circumference”.
无论点 C 位于优弧还是劣弧上，该定理都成立，并且它经常与其他定理结合出现在多步骤题目中。在爱德思 IGCSE 试卷中，你通常需要写出理由：“圆心角等于圆周角的两倍”。
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If ∠AOB = 70°, then ∠ACB = 35°.
若 ∠AOB = 70°，则 ∠ACB = 35°。
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The two angles must stand on the same arc AB.
这两个角必须对应同一条弧 AB。
∠AOB = 2 × ∠ACB
4. Angles in the Same Segment | 同弧上的圆周角
If two angles stand on the same chord and their vertices lie on the same side of the chord, then the two angles are equal. For points A, B, P and Q on a circle, if ∠APB and ∠AQB both stand on chord AB, then ∠APB = ∠AQB.
如果两个圆周角对应同一条弦，且它们的顶点位于弦的同一侧，那么这两个角相等。对于圆上的点 A、B、P、Q，如果 ∠APB 和 ∠AQB 都对应弦 AB，则 ∠APB = ∠AQB。
This is called the “angles in the same segment are equal” theorem. It is particularly useful when a diagram contains multiple points on the same circle and you need to transfer an angle from one position to another.
这被称为“同弧上的圆周角相等”定理。当图中圆上有多个点，并且你需要把一个角从一个位置转移到另一个位置时，这个定理特别有用。
Be careful: the vertices P and Q must lie on the same arc of chord AB. If they lie on opposite arcs, the angles are supplementary rather than equal.
请注意：顶点 P 和 Q 必须位于弦 AB 的同一侧弧上。如果它们位于相对的弧上，那么这两个角互补（和为 180°）而不是相等。
5. Cyclic Quadrilaterals | 圆内接四边形
A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the circumference of a circle. The most important property is that opposite angles add up to 180°. If A, B, C, D lie on a circle in that order, then ∠A + ∠C = 180° and ∠B + ∠D = 180°.
圆内接四边形是指四个顶点都在同一个圆上的四边形。最重要的性质是：对角互补，即 ∠A + ∠C = 180°，∠B + ∠D = 180°。
Another useful property is that the exterior angle of a cyclic quadrilateral equals the opposite interior angle. For example, if side AB is extended to E, then ∠CBE = ∠ADC. This is often tested in the Edexcel IGCSE papers as part of a two-mark “find the angle” question.
另一个有用的性质是：圆内接四边形的外角等于其内对角。例如，如果边 AB 延长到 E，则 ∠CBE = ∠ADC。这一性质常出现在爱德思 IGCSE 试卷中，作为两分值的角度求解题。
Opposite angles of a cyclic quadrilateral sum to 180°.
圆内接四边形对角之和为 180°。
6. Tangent and Radius | 切线与半径
A tangent to a circle is perpendicular to the radius drawn to the point of contact. If a line touches a circle at point T, and O is the centre, then OT ⊥ tangent at T, so the angle between OT and the tangent line is 90°.
圆的切线垂直于过切点的半径。如果一条直线与圆相切于点 T，O 是圆心，则 OT ⊥ 切线，即 OT 与切线之间的夹角为 90°。
This theorem is often used together with the tangent length theorem: if PA and PB are two tangents drawn from an external point P to a circle, then PA = PB. The two tangents from the same external point are always equal in length.
这个定理常与切线长定理一起使用：若从圆外一点 P 作圆的两条切线 PA 和 PB，则 PA = PB。从同一点出发的两条切线长度始终相等。
Radius ⊥ Tangent at the point of contact; PA = PB.
半径 ⊥ 切线（在切点处）；PA = PB。
7. The 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. If the tangent touches the circle at T, and chord TA is drawn, then the angle between the tangent and TA equals the angle subtended by TA at any point on the opposite arc.
弦切角定理指出：切线与过切点的弦所成的角，等于该弦在另一侧弧上任意一点所对的圆周角。如果切线在点 T 与圆相切，并作弦 TA，则切线与 TA 的夹角等于 TA 在另一侧弧上任意一点所对的角。
In the Edexcel IGCSE syllabus, this theorem is often called the “tangent-chord theorem”. It is one of the most powerful tools in circle geometry because it directly connects a tangent angle to an angle inside the circle.
在爱德思 IGCSE 课程中，这个定理通常被称为“切线-弦定理”。它是圆几何中最有力的工具之一，因为它直接把切线的角与圆内部的角联系起来。
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The angle must be between the tangent and the chord.
该角必须是切线与弦之间的夹角。
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The equal angle is found in the segment on the opposite side of the chord.
相等的角位于弦的另一侧弓形中。
8. Chord Properties | 弦的性质
There are two important chord properties you should know for the IGCSE examination. First, the perpendicular from the centre of a circle to a chord bisects the chord. If O is the centre, AB is a chord, and M is the point where the perpendicular from O meets AB, then AM = MB.
在 IGCSE 考试中，你需要掌握两条重要的弦性质。第一，圆心到弦的垂线平分这条弦。如果 O 为圆心，AB 为弦，M 是从 O 向 AB 作垂线的垂足，则 AM = MB。
Second, the perpendicular bisector of a chord always passes through the centre of the circle. These properties are often needed when solving problems involving radius, chord length and distance from the centre, especially combined with Pythagoras’ theorem.
第二，弦的垂直平分线必定经过圆心。在解决涉及半径、弦长和圆心到弦距离的问题时，这些性质经常与勾股定理结合使用。
For example, if a chord of length 10 cm is at a distance of 12 cm from the centre, the radius can be found using Pythagoras: r = √(5² + 12²) = √(25 + 144) = √169 = 13 cm.
例如，如果一条长度为 10 cm 的弦距圆心 12 cm，则可以利用勾股定理求半径：r = √(5² + 12²) = √(25 + 144) = √169 = 13 cm。
9. Worked Example 1 | 例题一：圆心角与圆周角
A, B and C are points on a circle with centre O. The angle AOB = 56°. Find angle ACB and give a reason for your answer.
A、B、C 是圆心为 O 的圆上的点。∠AOB = 56°。求 ∠ACB 并说明理由。
Solution / 解答: Since O is the centre, ∠AOB is the angle at the centre standing on arc AB. Angle ACB is the angle at the circumference standing on the same arc AB. Therefore ∠ACB = ½ × 56° = 28°.
因为 O 是圆心，∠AOB 是对应弧 AB 的圆心角；∠ACB 是对应同一条弧 AB 的圆周角。所以 ∠ACB = ½ × 56° = 28°。
∠ACB = 28°
Reason: the angle at the centre is twice the angle at the circumference standing on the same arc.
理由：圆心角是同一弧所对圆周角的两倍。
10. Worked Example 2 | 例题二：半圆与圆内接四边形
A, B, C and D are points on a circle. AB is a diameter. Angle BAC = 34°. Find angle ADC.
A、B、C、D 是圆上的点，AB 是直径。∠BAC = 34°。求 ∠ADC。
Step 1: Since AB is a diameter, the angle in a semicircle gives ∠ACB = 90°. In triangle ABC, the angle sum is 180°, so ∠ABC = 180° − 90° − 34° = 56°.
第一步：因为 AB 是直径，根据半圆中的角定理，∠ACB = 90°。在三角形 ABC 中，内角和为 180°，所以 ∠ABC = 180° − 90° − 34° = 56°。
Step 2: ABCD is a cyclic quadrilateral because all four points lie on the same circle. Therefore opposite angles sum to 180°: ∠ABC + ∠ADC = 180°. Hence ∠ADC = 180° − 56° = 124°.
第二步：因为四个点都在同一个圆上，ABCD 是圆内接四边形，所以对角互补：∠ABC + ∠ADC = 180°。因此 ∠ADC = 180° − 56° = 124°。
∠ADC = 124°
This example shows how two different circle theorems must often be used in sequence to reach the final answer.
这个例子说明，在解决综合题时，往往需要先后运用两个不同的圆的定理才能得出最终答案。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many students lose marks in circle theorem questions by forgetting to give a reason or by quoting the wrong theorem. In the Edexcel IGCSE mark scheme, the angle calculation and the reason are usually worth separate marks, so always write both.
许多学生在圆的定理题目中失分，是因为忘记写理由，或者引用了错误的定理。在爱德思 IGCSE 评分标准中，角度计算和理由通常分别占分，所以一定要同时写出两者。
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Always check that the angle at the centre and the angle at the circumference stand on the same arc.
始终检查圆心角与圆周角是否对应同一条弧。
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Do not assume a line is a diameter unless it is stated or clearly passes through the centre.
不要想当然地认为某条线段是直径，除非题目明确说明或它明显经过圆心。
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When a tangent appears, immediately mark the 90° angle between the tangent and the radius.
当图中出现切线时，立刻标出切线与半径之间的 90° 角。
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In multi-step problems, write down every angle you find as you go; intermediate angles are often worth credit.
在多步骤问题中，逐步写下一个一个求出的角度；中间角度往往也能得分。
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Learn the exact wording of each theorem so you can quote it confidently in the exam.
记住每个定理的准确表述，以便在考试中自信地引用。
12. Summary Table | 定理总结表
The table below lists all the key circle theorems you need for the Edexcel IGCSE Mathematics examination.
下表列出了爱德思 IGCSE 数学考试中你需要掌握的所有重要圆的定理。
| Theorem / 定理 | Statement / 结论 |
| Angle in a semicircle / 半圆中的角 | ∠ACB = 90° if AB is a diameter / 若 AB 为直径，则 ∠ACB = 90° |
| Angle at the centre / 圆心角 | ∠AOB = 2 × ∠ACB / 圆心角是圆周角的两倍 |
| Same segment / 同弧上的圆周角 | ∠APB = ∠AQB / 同弧上的圆周角相等 |
| Cyclic quadrilateral / 圆内接四边形 | Opposite angles sum to 180° / 对角互补 |
| Tangent and radius / 切线与半径 | Radius ⊥ Tangent / 半径垂直于切线 |
| Alternate segment /
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