📚 IGCSE Maths G-3 Practice Book 310: Circle Theorems Masterclass | IGCSE 数学 G-3 练习册 310：圆定理精讲精练
In this G-3 Practice Book 310 article, we work through the circle theorems that regularly appear in IGCSE Mathematics exams. You will learn the key angle facts, the eight core circle theorems, and how to apply them with clear proof-style reasoning.
在 G-3 练习册 310 的内容中，我们将系统梳理 IGCSE 数学考试中频繁出现的圆定理。你将掌握基本角度事实、八个核心圆定理，以及如何用清晰的推理方式应用它们。
1. Angle Facts You Must Know | 你必须掌握的角度事实
Before using circle theorems, make sure you are fluent with angle facts on straight lines, around points, and between parallel lines. These rules are the building blocks for many circle-theorem deductions.
在使用圆定理之前，请确保你熟练掌握直线、点周围以及平行线之间的角度事实。这些规则是许多圆定理推理的基础。
Angles on a straight line add up to 180°, and angles around a point add up to 360°. Vertically opposite angles are equal.
直线上的角之和为 180°，点周围的角之和为 360°。对顶角相等。
When a transversal crosses two parallel lines, alternate angles are equal, corresponding angles are equal, and co-interior angles add up to 180°.
当一条截线穿过两条平行线时，内错角相等，同位角相等，同旁内角之和为 180°。
- Straight line: ∠A + ∠B = 180°
- 直线：∠A + ∠B = 180°
- Around a point: ∠A + ∠B + ∠C + ∠D = 360°
- 点周围：∠A + ∠B + ∠C + ∠D = 360°
- Vertically opposite: ∠A = ∠C
- 对顶角：∠A = ∠C
Always mark these basic angles clearly on your diagram before applying any circle theorem.
在应用任何圆定理之前，请先在图上清楚地标出这些基本角度。
2. Circle Notation and Key Terms | 圆的记法与关键术语
A circle has a centre O, a radius r, a chord, a tangent, an arc, and a sector. You need to name points and angles precisely in diagrams.
圆有圆心 O、半径 r、弦、切线、弧和扇形。你需要在图形中准确命名点和角。
A tangent is a straight line that touches the circle at exactly one point. A chord joins two points on the circumference. A diameter is a special chord that passes through the centre.
切线是与圆只有一个交点的直线。弦连接圆周上的两个点。直径是经过圆心的特殊弦。
In IGCSE questions, the centre is usually labelled O, points on the circumference are labelled with capital letters such as A, B, P, Q, and the tangent is often drawn from an external point T.
在 IGCSE 题目中，圆心通常标记为 O，圆周上的点用大写字母如 A、B、P、Q 标记，切线通常从外部点 T 画出。
Understanding this notation helps you follow the wording of the question and write clear angle statements.
理解这些记法有助于你读懂题目表述，并写出清晰的角度表达式。
3. Theorem 1: Angle at the Centre Is Twice the Angle at the Circumference | 定理 1：圆心角是圆周角的两倍
If an arc subtends an angle at the centre O and an angle at a point P on the circumference, then the central angle is twice the angle at the circumference.
如果同一段弧在圆心 O 处和圆周上点 P 处分别形成角，那么圆心角是圆周角的两倍。
∠AOB = 2 × ∠APB
This theorem works even if P lies on the major arc, as long as the angles are subtended by the same arc AB.
只要两个角是由同一段弧 AB 所对，即使点 P 在优弧上，该定理仍然成立。
The result follows from the isosceles triangles formed by radii OA, OB, and OP. Since OA = OP, triangle OAP is isosceles, and its base angles are equal.
这个结论来源于半径 OA、OB 和 OP 形成的等腰三角形。因为 OA = OP，三角形 OAP 是等腰三角形，所以底角相等。
Use this theorem when you are given a central angle and need to find an angle on the circumference, or the reverse.
当你已知圆心角并需要求圆周角，或已知圆周角求圆心角时，使用这个定理。
4. Theorem 2: Angles in the Same Segment Are Equal | 定理 2：同弧上的圆周角相等
All angles in the same segment, standing on the same chord, are equal. In other words, if P and Q lie on the same side of chord AB, then ∠APB = ∠AQB.
同一弓形内、立于同一弦上的所有圆周角都相等。也就是说，如果点 P 和 Q 在弦 AB 的同侧，那么 ∠APB = ∠AQB。
∠APB = ∠AQB
This result is extremely useful when a diagram contains several triangles sharing a side inside a circle.
当图形包含圆内几个共享同一条边的三角形时，这个结论非常有用。
It follows directly from Theorem 1: both ∠APB and ∠AQB are half of the same central angle ∠AOB.
该定理直接由定理 1 推出：∠APB 和 ∠AQB 都是同一个圆心角 ∠AOB 的一半。
Be careful to check that the angles are subtended by the same chord and lie in the same segment, not opposite segments.
注意检查这些角是由同一条弦所对，并且位于同一弓形内，而不是相对的弓形中。
5. Theorem 3: The Angle in a Semicircle Is a Right Angle | 定理 3：半圆内的圆周角是直角
If AB is a diameter, then any angle subtended by AB at the circumference is 90°. This is a special case of Theorem 1 because the central angle is 180°.
如果 AB 是直径，那么圆周上任意一点对 AB 所张的角都是 90°。这是定理 1 的特例，因为此时圆心角为 180°。
∠APB = 90°
Use this theorem whenever you see a diameter in a circle question; it often gives the right angle needed for Pythagoras or trigonometry.
只要在圆的问题中看到直径，就可以使用这个定理；它通常提供毕达哥拉斯或三角函数所需的直角。
For example, if AB is a diameter and ∠PAB = 35°, then ∠ABP = 55° because the three angles in triangle APB add up to 180°.
例如，如果 AB 是直径且 ∠PAB = 35°，那么 ∠ABP = 55°，因为三角形 APB 的三个内角之和为 180°。
This theorem also helps identify the hypotenuse of a right-angled triangle inside a circle.
该定理还有助于识别圆内直角三角形的斜边。
6. Theorem 4: Cyclic Quadrilaterals Have Opposite Angles Adding to 180° | 定理 4：圆内接四边形对角互补
A cyclic quadrilateral is a four-sided figure with all four vertices on the circumference. The sum of each pair of opposite angles is 180°.
圆内接四边形是四个顶点都在圆周上的四边形。每一组对角之和均为 180°。
∠A + ∠C = 180° and ∠B + ∠D = 180°
The exterior angle at a vertex is equal to the interior opposite angle. This gives a quick shortcut in many exam questions.
顶点处的外角等于其内对角。这在许多考题中是一个快捷方法。
This rule is often tested with simultaneous equations when one pair of opposite angles is written as algebraic expressions, such as 2x + 30 and x + 50.
当一组对角写成代数式时，例如 2x + 30 和 x + 50，该规则常与方程求解一起考查。
Set the pair of opposite angles equal to 180°, solve for x, and then substitute back to find all angles.
令这组对角之和等于 180°，解出 x，再代回求出所有角。
7. Theorem 5: A Tangent Is Perpendicular to the Radius | 定理 5：切线与半径垂直
A tangent to a circle is perpendicular to the radius drawn to the point
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