📚 IGCSE Circle Theorems: Angle Rules You Must Know | IGCSE 圆定理：你必须掌握的角规则
In IGCSE Mathematics, circle theorems are a key part of the geometry section of the syllabus. They describe the angle and length relationships between points, chords, tangents and the centre of a circle, and they regularly appear in Paper 2 (structured questions with short steps) and Paper 4 (extended, multi-step problems). Most of the marks come from 'find the angle' and 'show that' questions, where you must chain two or three theorems together and justify every single step with a standard reason.
在 IGCSE 数学中，圆定理是大纲几何部分的核心内容。它们描述了圆上的点、弦、切线与圆心之间的角和长度关系，并且经常出现在 Paper 2（步骤较短的结构化题）与 Paper 4（多步骤拓展题）中。大部分分数来自 'find the angle' 和 'show that' 类题目，你需要把两三个定理串联起来，并为每一步给出标准理由。
This revision guide covers every circle theorem on the specification, the exact wording examiners expect, the worked examples that appear most often, and the mistakes that quietly cost marks even when the geometry is correct.
本复习指南覆盖考纲中的全部圆定理、考官期待的标准表述、最常出现的例题，以及那些"几何想对了却白白丢分"的常见错误。
1. Circle Theorem Basics | 圆定理基础
A circle theorem is a geometric rule that connects angles, lines and points on the circumference. In IGCSE you do not need to prove every theorem, but you must recognise the diagram pattern quickly and write down the correct angle or length relationship together with its reason. Key elements of every circle diagram include the centre O, the radius r, chords, tangents, arcs, segments, sectors and the circumference itself.
圆定理是联系圆周上的角、线和点的几何规则。在 IGCSE 中你不必证明每一个定理，但必须快速识别图形模式，写出正确的角或长度关系，并附上理由。每个圆图的关键元素包括圆心 O、半径 r、弦、切线、弧、弓形、扇形以及圆周本身。
Circle theorem questions almost always follow one of three shapes: a single-step question ("find x, giving a reason"), a two-step chain (for example angle at the centre, then a cyclic quadrilateral), or a proof-style 'show that' question where the final answer is given and all the marks are for the logical chain.
圆定理题目几乎总是三种形态之一：单步题（"求 x，并给出理由"）、两步串联题（例如先用圆心角定理，再用圆内接四边形），以及"show that"证明题——最终答案已经给出，所有分数都在逻辑链上。
What the examiner rewards is not the arithmetic. It is the pattern recognition plus the named theorem. A correct number with no reason usually scores half marks at best; a correct number with the exact standard reason scores full marks.
考官给分的重点不是算术，而是图形识别加上定理名称。只写对数字不写理由，通常最多拿一半分；写对数字并配上标准理由，才能拿满分。
2. Key Vocabulary and Diagram Labels | 关键术语与图形标注
Before the theorems make sense, the vocabulary must be exact. A chord is a straight line joining two points on the circumference; a tangent touches the circle at exactly one point and never crosses it; a segment is the region between a chord and an arc; a sector is the region between two radii and an arc. The verb "subtend" simply means "is opposite to" - an angle subtends an arc or a chord when its two arms end at the two endpoints of that arc or chord.
在理解定理之前，术语必须先弄清楚。弦是连接圆周上两点的直线；切线只与圆相交于一点，且不穿过圆；弓形是弦与弧之间的区域；扇形是两条半径与一段弧之间的区域。动词 "subtend"（对）的意思就是"位于某物的对面"——当一个角的两条边落在某段弧或某条弦的两个端点上时，就说这个角"对"这段弧或这条弦。
| Term | 中文 | Meaning in a diagram |
| Centre O | 圆心 O | The fixed point equidistant from every point on the circle. |
| Radius r | 半径 r | Any line from O to the circumference; all radii are equal. |
| Diameter d | 直径 d | A chord through the centre, d = 2r. |
| Chord | 弦 | Straight line joining two points on the circumference. |
| Tangent | 切线 | Line touching the circle at exactly one point (the point of contact). |
| Arc | 弧 | Part of the circumference; minor (short) or major (long). |
| Segment | 弓形 | Region cut off by a chord. |
| Sector | 扇形 | Region between two radii and an arc. |
| Cyclic quadrilateral | 圆内接四边形 | A four-sided shape with all four vertices on the circle. |
| Subtend | 对（弧/弦） | The angle whose two arms end at the ends of that arc or chord. |
Whenever a diagram shows two radii meeting at the centre, you should immediately see an isosceles triangle, because both radii are equal. This single observation unlocks many otherwise difficult questions.
只要图中出现两条半径交于圆心，你就应该立刻看到一个等腰三角形，因为两条半径相等。仅凭这一点，就能解开很多看似困难的题目。
3. Angle at the Centre Theorem | 圆心角定理
The angle subtended at the centre of a circle by an arc is twice the angle subtended at the circumference by the same arc. In a diagram, if A and B lie on the circumference, O is the centre, and P is another point on the circumference, then the central angle ∠AOB and the inscribed angle ∠APB are related by the rule below.
同一段弧所对的圆心角，是同一段弧所对圆周角的两倍。在图中，如果 A、B 在圆周上，O 是圆心，P 是圆周上的另一点，那么圆心角 ∠AOB 与圆周角 ∠APB 满足下面的关系。
∠AOB = 2 × ∠APB
∠APB = ½ × ∠AOB
The reason you write is: "the angle at the centre is twice the angle at the circumference subtended by the same arc". Never write only "circle theorem" - that earns nothing.
你要写的理由是："the angle at the centre is twice the angle at the circumference subtended by the same arc"（同弧所对的圆心角是圆周角的两倍）。千万不要只写 "circle theorem"，那样拿不到分。
Worked mini-example: if ∠APB = 35°, then ∠AOB = 2 × 35° = 70°. If instead the question gives ∠AOB = 118°, then ∠APB = ½ × 118° = 59°.
小例题：若 ∠APB = 35°，则 ∠AOB = 2 × 35° = 70°。若题目给出 ∠AOB = 118°，则 ∠APB = ½ × 118° = 59°。
ngle at the centre is 360° − 250° = 110°, and the angle at the circumference is ½ × 110° = 55°. Always ask which arc the two angles actually subtend.
要留意优角（大于 180°）的情况。如果 P 在劣弧上，而题目给出的圆心角是优角（例如 250°），那么圆心处的非优角为 360° − 250° = 110°，对应的圆周角为 ½ × 110° = 55°。始终要先判断这两个角究竟对的是哪一段弧。
4. Angle in a Semicircle | 半圆上的圆周角
If AB is a diameter and P is any other point on the circumference, then ∠ straight angle of 180° at the centre, so the angle at the circumference is ½ × 180° = 90°.
如果 AB 是直径，P 是圆周上任意另一点，那么 ∠APB = 90°。这实际上是圆心角定理的特例：直径在圆心处形成 180° 的平角，因此圆周角为 ½ × 180° = 90°。
AB = diameter ⟹ ∠APB = 90°
Standard reason: "the angle in a semicircle is a right angle", or "the angle subtended by a diameter is 90°".
标准理由："the angle in a semicircle is a right angle"，或 "the angle subtended by a diameter is 90°"（直径所对的圆周角是直角）。
The converse is equally examinable: if you are told that ∠ t be a diameter, so the centre O lies on AB and OA = OB = OP. This is a favourite trick in 'show that O lies on AB' questions.
逆定理同样常考：如果已知 ∠APB = 90°，那么 AB 必为直径，因此圆心 O 在 AB 上，且 OA = OB = OP。这是 "show that O lies on AB" 类题目的常见套路。
ined with a given chord length and radius, this links circle theorems to your length work.
由于三角形 APB 是直角三角形，你还可以用勾股定理：AB² = AP² + PB²。结合已知的弦长和半径，这就把圆定理与长度计算联系了起来。
5. Angles in the Same Segment | 同弓形（同弧）圆周角相等
on the same side of the chord, are equal. So if P and Q are both on the major arc AB, then ∠APB = ∠AQB. The reason is: "angles in the same segment are equal" or "angles subtended by the same arc are equal".
圆周上位于弦同侧、对同一段弧的角相等。因此若 P、Q 都在优弧 AB 上，则 ∠APB = ∠AQB。理由是："angles in the same segment are equal"，或 "angles subtended by the same arc are equal"（同弧所对的圆周角相等）。
∠APB = ∠AQB (same segment)
This theorem is the fastest way to move an angle from one part of a crowded diagram to another, and it is often the hidden step in a longer question. If you can spot two angles standing on the same chord with their vertices on the same arc, you have a free equation.
这个定理是把角从密密麻麻的图形一处"搬运"到另一处的最快方法，也常常是长题中被隐藏的一步。只要你能看出两个角的顶点在同一段弧上、且对着同一条弦，你就白得一个等式。
A useful companion fact: if two points are on opposite arcs, the angles are supplementary (they add to 180°) because the four points form a cyclic quadrilateral. So the same diagram can be read with either theorem - choose the one that fits the given data.
一个有用的配套结论：如果两点分别位于两段相对的弧上，那么这两个角互补（和为 180°），因为四个点构成圆内接四边形。因此同一张图可以用两个定理来读——选择与已知条件相符的那一个。
6. Cyclic Quadrilateral Theorem | 圆内接四边形定理
If all four vertices A, B, C, D of a quadrilateral lie on a circle, then each pair of opposite angles sums to 180°. The reason is: "opposite angles of a cyclic quadrilateral sum to 180°".
如果四边形 ABCD 的四个顶点都在同一个圆上，那么每一对对角之和为 180°。理由是："opposite angles of a cyclic quadrilateral sum to 180°"（圆内接四边形对角互补）。
∠A + ∠C = 180°
∠B + ∠D = 180°
uadrilateral equals the interior opposite angle. If side AB is extended to a point E, then ∠CBE = ∠ADC. The reason is: "exterior angle of a cyclic quadrilateral equals the interior opposite angle".
还有一条密切相关、能省时间的规则：圆内接四边形的外角等于内对角。若边 AB 延长到点 E，则 ∠CBE = ∠ADC。理由是："exterior angle of a cyclic quadrilateral equals the interior opposite angle"（圆内接四边形的外角等于内对角）。
To use this theorem you must first prove or accept that the shape is cyclic. Look for four points on a circle, four points joined on the circumference, or a statement such as "A, B, C and D lie on the circle". Never assume a quadrilateral is cyclic just because it looks round - the vertices must actually be on the circumference.
使用这个定理前，必须先证明或确认该四边形是圆内接的。注意四个点在圆上、四个点在圆周上相连，或题目中 "A, B, C and D lie on the circle" 这类表述。不要因为图形看起来"很圆"就假定它是圆内接四边形——顶点必须确实在圆周上。
7. Alternate Segment Theorem | 弦切角定理
The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. In practice: the tangent touches the circle at A, the chord is AB, and C is a point on the circle on the other side of AB; then the angle between the tangent and AB equals ∠ACB.
切线与弦在切点处所夹的角，等于另一个弓形（交错弓形）中的圆周角。具体来说：切线与圆相切于 A，弦为 AB，C 是位于 AB 另一侧圆周上的一点，那么切线与 AB 所夹的角等于 ∠ACB。
angle (tangent, chord AB) = ∠ACB
The reason to write is: "alternate segment theorem", or the fuller version "the angle between the tangent and the chord equals the angle in the alternate segment".
要写的理由是："alternate segment theorem"（弦切角定理），或更完整的 "the angle between the tangent and the chord equals the angle in the alternate segment"。
Two points decide whether you have used it correctly. First, the angle on the tangent side and the circle angle must be on opposite sides of the chord - that is what "alternate" means. Second, the chord must have one endpoint at the point of contact. If the tangent touches at A but your chord is CD with neither endpoint at A, the theorem does not apply directly.
判断是否用对，有两点：第一，切线上方的角与圆周角必须位于弦的两侧——这正是 "alternate"（交错）的含义；第二，弦必须有一个端点在切点上。如果切点是 A，而你的弦是 CD 且两端都不是 A，那么这个定理不能直接用。
This theorem is often combined with the tangent-radius rule in section 8 and with angles in a triangle summing to 180°, which is how most 'find the angle' questions are structured.
这个定理常与第 8 节的切线与半径规则、以及"三角形内角和 180°"配合使用，这也是大多数 "find the angle" 题目的结构方式。
8. Tangent-Radius Rule and Two Tangents | 切线与半径、两条切线
A tangent to a circle is perpendicular to the radius drawn to the point of contact. The reason is: "the tangent is perpendicular to the radius at the point of contact". This gives you a right angle, and therefore access to Pythagoras and to right-angle trigonometry.
圆的切线垂直于过切点的半径。理由是："the tangent is perpendicular to the radius at the point of contact"（切线垂直于过切点的半径）。这给你一个直角，从而可以用勾股定理和直角三角形三角比。
OT ⊥ TA at the point of contact T
² = OA². This is the standard route for finding a tangent length or a radius from lengths.
若 T 是切点，O 是圆心，A 是圆外一点，则三角形 OTA 在 T 处为直角，故 OT² + AT² = OA²。这是由长度求切线长
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