📚 Circle Theorems for IGCSE Mathematics | IGCSE 数学：圆定理精讲
Circle theorems are an essential part of the IGCSE Mathematics syllabus. They allow you to find unknown angles and prove geometric relationships inside circles without measuring. This revision guide covers the key theorems, worked applications and exam tips in a clear bilingual format.
圆定理是 IGCSE 数学大纲中的重要内容。它们帮助你无需测量就能求出圆中的未知角，并证明几何关系。本复习指南以清晰的中英双语形式讲解关键定理、应用方法和考试技巧。
1. What Are Circle Theorems? | 什么是圆定理？
Circle theorems are geometric rules that connect angles, chords, tangents and segments of a circle. In IGCSE Mathematics, they are used to calculate unknown angles and to prove relationships without relying on scale drawings. You must be able to state the theorem and apply it to diagrams.
圆定理是连接圆中角、弦、切线和弓形的几何规则。在 IGCSE 数学中，它们用于计算未知角，并在不依赖按比例绘图的情况下证明关系。你必须能够陈述定理并将其应用于图形中。
There are eight main circle theorems that appear in most IGCSE exams. Learning them as 'angle facts' will help you spot shortcuts in complex diagrams. Always give a reason when you write your working.
大多数 IGCSE 考试中会出现八个主要圆定理。把它们作为 '角度事实' 来学习，有助于你在复杂图形中发现捷径。书写解题过程时，一定要给出理由。
2. The Angle at the Centre Is Twice the Angle at the Circumference | 圆心角等于两倍圆周角
If an arc AB subtends an angle at the centre O and an angle at a point P on the remaining circumference, then the angle at the centre is twice the angle at the circumference. This theorem only applies when both angles stand on the same arc AB.
如果弧 AB 在圆心 O 处对着一个角，并且在圆周上另一点 P 处对着一个角，那么圆心角等于圆周角的两倍。该定理仅当两个角都对着同一条弧 AB 时成立。
∠AOB = 2 × ∠APB
This theorem is often used to find the centre angle when a circumference angle is known, or vice versa. In questions, look for an angle at the centre and another angle standing on the same chord.
该定理常用于已知圆周角求圆心角，或反向求解。在题目中，注意寻找一个圆心角和另一个对着同一条弦的角。
3. Angles in the Same Segment Are Equal | 同弓形内的圆周角相等
le are equal. In other words, if two points P and Q lie on the same side of chord AB, then ∠APB = ∠AQB.
同一条弦或弧在圆上同一弓形内所对的圆周角相等。也就是说，如果 P 和 Q 位于弦 AB 的同一侧，那么 ∠APB = ∠AQB。
This is extremely useful when a diagram contains several triangles built on the same base. You can transfer an angle from one triangle to another without proving congruence.
当图形中有多个以同一底边构造的三角形时，这一定理非常有用。你可以将一个角从一个三角形转移到另一个三角形，而无需证明全等。
4. The Angle in a Semicircle Is 90° | 半圆上的圆周角是直角
The angle subtended by a diameter at any point on the circumference is a right angle. Since a diameter is a straight line through the centre, it forms an angle of 180° at the centre, so by the first theorem the angle at the circumference is 90°.
直径在圆周上任一点所对的角是直角。由于直径是通过圆心的直线，它在圆心处形成 180° 角，因此根据第一个定理，圆周角为 90°。
If AB is a diameter, then ∠APB = 90°
This is often tested together with Pythagoras' theorem, because a triangle inscribed in a semicircle is always right-angled. If you see a diameter, immediately mark the right angle.
这常与毕达哥拉斯定理一起考查，因为内接于半圆的三角形总是直角三角形。如果你看到直径，立刻标出直角。
5. Cyclic Quadrilaterals: Opposite Angles Sum to 180° | 圆内接四边形对角互补
A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the circumference of a circle. In any cyclic quadrilateral, opposite angles add up to 180°.
圆内接四边形是指四个顶点都在圆周上的四边形。在任何圆内接四边形中，对角之和为 180°。
∠A + ∠C = 180° and ∠B + ∠D = 180°
This property is the converse of the fact that if a pair of opposite angles of a quadrilateral sums to 180°, the quadrilateral is cyclic. It is a common way to prove that four points lie on a circle.
该性质的逆命题是：如果四边形的一对对角之和为 180°，则该四边形是圆内接四边形。这是证明四点共圆的常用方法。
6. Tangent and Radius: A Tangent Meets the Radius at 90° | 切线与半径垂直
A tangent to a circle is a straight line that touches the circle at exactly one point. At the point of contact, the tangent is perpendicular to the radius drawn to that point.
圆的切线是一条与圆恰好只有一个交点的直线。在切点处，切线与过该点的半径垂直。
If PT is tangent at T and O is the centre, then ∠OTP = 90°
This theorem is often combined with Pythagoras' theorem to find the length of a tangent from an external point. Look for right-angled triangles involving the radius, tangent and distance from the centre to the external point.
该定理常与毕达哥拉斯定理结合，求圆外一点到圆的切线长度。注意寻找由半径、切线和圆心到圆外点的距离构成的直角三角形。
7. Tangents from an External Point Are Equal in Length | 圆外一点到圆的两条切线长相等
From an external point P, two tangents can be drawn to a circle, touching it at points T and U. The lengths PT and PU are equal.
从圆外一点 P 可以作两条切线，分别与圆相切于 T 和 U。切线长 PT 和 PU 相等。
PT = PU
Additionally, the line joining the external point to the centre bisects the angle between the two tangents. This fact is often used in problems involving kite-shaped tangents.
此外，圆外点与圆心的连线平分两条切线之间的夹角。这一事实常用于涉及筝形切线的问题中。
8. Alternate Segment Theorem | 交错弦切角定理
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.
交错弦切角定理指出，切线与经过切点的弦之间的角等于圆的交错弓形内的圆周角。
If AB is a chord and AT is tangent at A, then ∠BAT = ∠BCA
This is one of the hardest circle theorems for many students because the angle lies outside the circle. Draw the chord and tangent clearly, then shade the alternate segment to avoid confusion.
对许多学生来说，这是最难的圆定理之一，因为角位于圆的外部。清晰地画出弦和切线，然后给交错弓形涂上阴影，可以避免混淆。
9. Chord Properties: Perpendicular from Centre to Chord | 弦的性质：圆心到弦的垂线平分弦
If a line is drawn from the centre of a circle perpendicular to a chord, it bisects the chord. This means the perpendicular distance from the centre to the chord cuts the chord into two equal parts.
如果从圆心作一条垂直于弦的线，则该线平分弦。这意味着从圆心到弦的垂直距离将弦分成两等份。
If OM ⟂ AB, then AM = MB
This theorem also gives a useful right-angled triangle: the radius, half the chord and the perpendicular distance from the centre form a right triangle. It is widely used to find chord lengths or distances from the centre.
该定理还构造了一个有用的直角三角形：半径、弦的一半和圆心到弦的垂直距离构成直角三角形。它广泛用于求弦长或圆心到弦的距离。
10. Solving IGCSE Problems Using Circle Theorems | 运用圆定理解 IGCSE 考题
IGCSE questions often combine two or more circle theorems in one diagram. Start by marking all given angles and write down which theorem applies to each step.
IGCSE 考题常在一个图形中结合两个或多个圆定理。开始时先标出所有已知角，并写下每一步应用了哪个定理。
A typical strategy is to identify the centre, radii, tangents and cyclic quadrilaterals first. Then look for angles in the same segment, right angles in semicircles, or opposite angles summing to 180°.
一个典型策略是先识别圆心、半径、切线和圆内接四边形。然后寻找同弓形内的等角、半圆中的直角，或对角和为 180° 的情况。
Always state a reason such as 'angle at centre is twice angle at circumference' or 'alternate segment theorem'. This earns method marks even if the final angle is incorrect.
始终写出理由，例如 '圆心角等于两倍圆周角' 或 '交错弦切角定理'。即使最终角度错误，这也能获得方法分。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common mistake is applying the angle at the centre theorem to angles that do not stand on the same arc. Check that the two angles are subtended by the same chord or arc before using it.
一个常见错误是把圆心角定理应用于并非对着同一条弧的角。在使用前，请检查这两个角是否由同一条弦或弧所对。
Another frequent error is confusing the angle between tangent and radius with the alternate segment theorem. The tangent-radius angle is always 90°, while the alternate segment theorem involves a tangent and a chord.
另一个常见错误是混淆切线与半径的角和交错弦切角定理。切线与半径的角总是 90°，而交错弦切角定理涉及切线和弦。
Redraw the diagram large enough to label all points. If a question gives no diagram, sketch one neatly and mark the centre, tangents and chords clearly.
把图形重新画得足够大，以便标注所有点。如果题目没有给出图形，就整齐地画一个草图，并清楚地标出圆心、切线和弦。
12. Quick Summary Table | 圆定理速查表
Use this table as a rapid revision tool before your IGCSE exam. It summarises the eight core circle theorems and the key relation for each one.
在 IGCSE 考试前，将此表作为快速复习工具。它总结了八个核心圆定理及每个定理的关键关系。
| Theorem | Statement | Key relation |
|---|---|---|
| Angle at centre | Central angle is twice the angle at circumference | ∠O = 2 × ∠P |
| Angles in same segment | Angles on the same chord are equal | ∠P = ∠Q |
| Angle in semicircle | Angle in a semicircle is 90° | ∠APB = 90° |
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