📚 Circle Theorems: Angle Properties in IGCSE Mathematics | IGCSE 数学：圆定理与角的性质
Circle theorems are a core geometry topic in the IGCSE Mathematics syllabus, often introduced around the G-4 geometry section and page 100 of the student book. They appear regularly on both Core and Extended papers, and they provide a structured way to find unknown angles and lengths in circular diagrams. Because the reasoning is fixed and rule-based, this is one of the few geometry topics where careful memorisation of statements, combined with clear written reasons, almost guarantees full marks.
圆定理是 IGCSE 数学大纲中的核心几何主题，通常出现在学生用书 G-4 几何单元和第 100 页左右。它们经常出现在 Core 和 Extended 试卷中，为解决圆形图形中的未知角度和长度提供了系统方法。由于它的推理规则固定、有章可循，这是少数只要把定理条文背准、并把理由写清楚就几乎能拿满分的几何专题之一。
This article walks through every circle theorem you need, the exact wording of the reasons examiners expect, and the traps that cost candidates marks year after year.
本文将逐一梳理你需要掌握的每一条圆定理、考官期待的准确理由表述，以及年年让考生丢分的典型陷阱。
1. Circle Terminology | 圆的基本术语
A circle is the set of all points in a plane that are the same distance from a fixed point O, called the centre. The radius r is the distance from the centre to any point on the circle. A chord is a line segment joining two points on the circle, and a diameter is a chord that passes through the centre. A tangent is a straight line that touches the circle at exactly one point, while a secant cuts the circle at two points.
圆是平面上到定点 O（圆心）距离相等的所有点的集合。半径 r 是从圆心到圆上任意一点的距离。弦是连接圆上两点的线段，直径是经过圆心的弦。切线是与圆恰好相交于一点的直线，而割线与圆有两个交点。
An arc is part of the circumference. The major arc is longer than a semicircle, and the minor arc is shorter. A sector is the region enclosed by two radii and an arc, while a segment is the region enclosed by a chord and an arc. Knowing these terms is essential because circle theorem statements refer to chords, tangents, arcs and segments.
弧是圆周的一部分。优弧比半圆长，劣弧比半圆短。扇形是由两条半径和一段弧围成的区域，弓形是由一条弦和一段弧围成的区域。理解这些术语很重要，因为圆定理的表述会涉及弦、切线、弧和弓形。
Three further words appear constantly in exam wording. To subtend means to be opposite to: the chord AB subtends angle APB at the point P on the circumference. A cyclic quadrilateral is a four-sided shape whose four vertices all lie on a circle. The point of contact (or point of tangency) is the single point shared by a tangent and the circle. A point outside the circle from which tangents are drawn is called an external point.
还有三个词在试题中频繁出现。subend（对）意为"所对"：弦 AB 在圆周上的点 P 处所对的角是 ∠APB。圆内接四边形是指四个顶点都在同一个圆上的四边形。切点（point of contact / point of tangency）是切线与圆唯一共有的点。从圆外某一点向圆作切线，该点称为外点（external point）。
- Centre O, radius r, diameter d = 2r | 圆心 O、半径 r、直径 d = 2r
- Chord, tangent, secant | 弦、切线、割线
- Minor arc / major arc, minor sector / major sector | 劣弧 / 优弧、劣扇形 / 优扇形
- Segment, cyclic quadrilateral, point of contact | 弓形、圆内接四边形、切点
2. Theorem 1: Angle at the Centre | 定理 1：圆心角定理
The first and most heavily used theorem states that the angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circumference.
第一条也是使用最频繁的定理：一段弧在圆心处所对的角，等于它在圆周上任意一点所对的角的两倍。
∠AOB = 2 × ∠APB
Here A and B are the endpoints of the arc, O is the centre and P is any point on the major arc AB. If the angle at the centre is 130°, the angle at the circumference is 65°. Notice the direction of the reasoning matters: when the centre angle is known, halve it; when the circumference angle is known, double it.
其中 A、B 是这段弧的两个端点，O 是圆心，P 是优弧 AB 上的任意一点。若圆心角为 130°，则圆周角为 65°。注意推理方向：已知圆心角就除以 2，已知圆周角就乘以 2。
− 260° = 100°, and the angle at the circumference is 50°. Always check whether you are looking at the angle containing the arc, not the reflex angle by accident.
一个常见的特殊情况是圆心的优角（反射角）。若图中标出的圆心角是优角，例如 260°，则非优角为 360° − 260° = 100°，圆周角为 50°。务必确认你用的是对着那段弧的角，而不是不小心用了优角。
Exam reason to write: "angle at centre is twice angle at circumference".
考试中应写的理由："圆心角是圆周角的两倍"（angle at centre is twice angle at circumference）。
3. Theorem 2: Angle in a Semicircle | 定理 2：半圆所对的圆周角
e centre is a straight angle of 180°, so the angle at the circumference is 180° ÷ 2 = 90°.
半圆所对的圆周角是直角。这实际上是定理 1 的特例：若弧 AB 是半圆，则圆心角为平角 180°，于是圆周角为 180° ÷ 2 = 90°。
If AB is a diameter, then ∠APB = 90°
This theorem is the reason a triangle built on a diameter is always right-angled at the third vertex. In exam diagrams, a right angle sitting on the circumference is a strong hint that the opposite side is a diameter, and vice versa. The converse is also true and examinable: if the angle subtended by a chord at the circumference is 90°, that chord must be a diameter.
正因为这条定理，以直径为一边、第三点在圆上的三角形，在第三点处必为直角。在试题图中，若看到圆周上的直角，就强烈提示它所对的边是直径；反过来也一样。逆定理同样可考：若某条弦在圆周上所对的角为 90°，那么这条弦一定是直径。
Exam reason to write: "angle in a semicircle is 90°" or "angle in a semicircle" (abbreviated in mark schemes).
考试中应写的理由："半圆上的圆周角是 90°"（angle in a semicircle is 90°）。
4. Theorem 3: Angles in the Same Segment | 定理 3：同弧上的圆周角
l. If two points P and Q lie on the same arc AB and on the same side of the chord AB, then ∠APB = ∠AQB.
同一弓形内的圆周角相等。若 P、Q 两点位于同一段弧 AB 上、且在弦 AB 的同一侧，则 ∠APB = ∠AQB。
∠APB = ∠AQB (same segment)
The logic behind it is the centre angle theorem applied twice: both angles equal half of the same angle at the centre. This is the theorem that turns awkward, scattered angles in a diagram into equal angles, and it is often the key step that unlocks a multi-mark question.
它的原理是把圆心角定理用两次：两个角都等于同一个圆心角的一半。这条定理能把图中零散、看似无关的角变成相等的角，往往是解开多分大题的关键一步。
Watch the wording carefully. If two angles stand on the same chord but on opposite sides of it, they are not equal; instead they are supplementary (they sum to 180°) because the quadrilateral formed is cyclic. Always check which side of the chord the points lie on.
要特别小心措辞。若两个角所对的弦相同但在弦的两侧，它们并不相等，而是互补（和为 180°），因为此时构成的四边形是圆内接四边形。务必确认这些点在弦的哪一侧。
Exam reason to write: "angles in the same segment are equal".
考试中应写的理由："同弧上的圆周角相等"（angles in the same segment）。
5. Theorem 4: Cyclic Quadrilaterals | 定理 4：圆内接四边形
A cyclic quadrilateral is a quadrilateral whose four vertices lie on a circle. Its opposite angles are supplementary: each pair of opposite angles adds up to 180°.
圆内接四边形是四个顶点都在同一个圆上的四边形。它的对角互补：每一组对角之和为 180°。
∠A + ∠C = 180° and ∠B + ∠D = 180°
There is a useful extension called the exterior angle property: if one side of a cyclic quadrilateral is extended, the exterior angle formed equals the interior opposite angle. This saves time when a diagram extends a side rather than marking the interior angle.
它还有一个实用的推论——外角性质：若延长圆内接四边形的一边，所得外角等于内对角。当题目图形延长了某条边而不是标出内角时，用它可以省下不少时间。
Two warnings. First, "opposite" means opposite, not adjacent: adjacent angles of a cyclic quadrilateral are generally not supplementary. Second, the 180° rule only applies when all four vertices are genuinely on the circle; a quadrilateral that merely looks circular does not qualify unless the diagram or question says so.
两点提醒。第一，"对角"就是对角的两个角，不是相邻角：圆内接四边形的相邻角一般并不互补。第二，180° 的规则只适用于四个顶点确实都在圆上的情形；仅仅"看起来像"圆内接的四边形，除非题目明确说明，否则不能使用。
Exam reason to write: "opposite angles of a cyclic quadrilateral sum to 180°".
考试中应写的理由："圆内接四边形对角互补"（opposite angles of a cyclic quadrilateral）。
6. Theorem 5: The Alternate Segment Theorem | 定理 5：弦切角定理
The angle between a tangent and a chord at the point of contact equals the angle in the alternate segment. In other words, the angle the chord makes with the tangent is equal to the inscribed angle standing on the same chord, measured on the opposite side of the chord.
切线与弦在切点处所成的角，等于该弦所对另一弓形内的圆周角。换句话说，弦与切线的夹角，等于同一弦所对、且在弦另一侧的圆周角。
∠(tangent, chord AB) = ∠APB in the alternate segment
There are two angles between the tangent and the chord, one on each side, and there are two arcs; the theorem links each one to its "alternate" partner. A reliable way to apply it is to identify the chord first, then look for the triangle that sits in the segment cut off by that chord.
切线与弦之间有两个夹角，分别位于两侧；同时也有两段弧。定理把每一个夹角与它"对面"的那段弧所对的圆周角对应起来。稳妥的做法是：先确定是哪条弦，再找由这条弦切出的弓形中那个三角形。
This theorem appears less often than the first four but carries disproportionate marks in Extended papers, usually in a two-part question where you must first find an angle using tangents and radii, then apply the alternate segment result. Students who skip it frequently lose four or five marks in one question.
这条定理出现频率低于前四条，但在 Extended 试卷中往往分值很高，通常是一道分两小问的题：先用切线与半径求出一个角，再套用弦切角定理。跳过它的学生常常一题就丢掉四五分。
Exam reason to write: "alternate segment theorem".
考试中应写的理由："弦切角定理"（alternate segment theorem）。
7. Theorem 6: Tangents and Radii | 定理 6：切线与半径
A tangent to a circle is perpendicular to the radius drawn to the point of contact. This gives a right angle at every point of tangency, which is why tangent questions are almost always solved with Pythagoras' theorem or right-angled trigonometry.
圆的切线垂直于过切点的半径。这就在每个切点处给出一个直角，因此切线类题目几乎总要用勾股定理或直角三角形三角比来解。
∠(tangent, radius) = 90°
If a tangent of length t is drawn from an external point P at distance d from the centre to a circle of radius r, then the right-angled triangle formed gives:
若从圆外一点 P 向半径为 r 的圆作长为 t 的切线，P 到圆心的距离为 d，则所形成的直角三角形给出：
d² = r² + t²
A frequent exam twist is a tangent that meets a chord or another tangent, producing an isosceles triangle or a kite, because the two radii are equal. Whenever you see two radii drawn in the same diagram, look for an isosceles triangle and use base angles equal.
常见考法变形是：切线再与一条弦或另一条切线相交，由于两条半径相等，就会产生等腰三角形或筝形。只要图中出现两条半径，就要想到等腰三角形，用"两底角相等"。
Exam reason to write: "tangent is perpendicular to radius at point of contact".
考试中应写的理由："切线垂直于过切点的半径"（tangent perpendicular to radius）。
8. Theorem 7: Two Tangents from an External Point | 定理 7：外点引两条切线
t to a circle are equal in length. If PA and PB touch the circle at A and B, then PA = PB.
从圆外一点向圆所作的两条切线长度相等。若 PA、PB 分别切圆于 A、B，则 PA = PB。
PA = PB, ∠APO = ∠BPO, ∠AOP = ∠BOP
Because the radii OA and OB are equal, the radii are perpendicular to the tangents, and the line PO is common, triangles OAP and OBP are congruent (RHS). It follows that the line from the external point to the centre bisects the angle between the tangents and also bisects the angle AOB at the centre.
由于半径 OA = OB、半径与切线垂直、且 PO 为公共边，三角形 OAP 与 OBP 全等（RHS）。由此可知，从外点到圆心的连线平分两切线之间的夹角，也平分圆心的 ∠AOB。
This result is the backbone of "kite" problems and of questions involving a circle inscribed in a triangle, where the tangents to the incircle create equal segments on each side. Recognising congruent triangles here often converts a hard-looking question into two lines of working.
这条结论是"筝形"类问题以及三角形内切圆问题的主干：内切圆的切线在各边上截出相等的线段。认出这里的全等三角形，常常能把一道看起来很难的题压缩成两行运算。
Exam reason to write: "tangents from an external point are equal in length".
考试中应写的理由："从圆外一点引出的切线长相等"（tangents from an external point）。
9. Theorem 8: Perpendicular from the Centre to a Chord | 定理 8：垂径定理
The perpendicular drawn from the centre of a circle to a chord bisects the chord. Conversely, the line from the centre that bisects a chord is perpendicular to it.
从圆心向弦作垂线，垂足平分这条弦。反之，从圆心出发平分弦的直线垂直于该弦。
If OM ⊥ AB, then AM = MB
Combined with Pythagoras, this gives the workhorse formula for chord problems. If a chord of length 2a is at perpendicular distance x from the centre of a circle with radius r, then:
与勾股定理结合，就得到解决弦长问题的主力公式。若长为 2a 的弦到半径为 r 的圆心的垂直距离为 x，则：
r² = a² + x²
This relationship answers the three standard questions: find the radius given a chord and its distance, find the chord given radius and distance, or find the distance given radius and chord. Substitute carefully — remember that a is half the chord, not the whole chord.
这一关系式可以解决三类标准问题：已知弦和距离求半径、已知半径和距离求弦长、已知半径和弦长求距离。代入时要仔细——a 是弦长的一半，不是整条弦。
Exam reason to write: "perpendicular from centre to chord bisects the chord".
考试中应写的理由："垂直于弦的直径平分弦"（perpendicular from centre bisects chord）。
10. Theorem 9: Equal Chords and Equal Arcs | 定理 9：等弦、等弧与弦心距
Equal chords in a circle are equidistant from the centre, and equal chords subtend equal angles at the centre. The converses are also true: chords equidistant from the centre are equal, and chords subtending equal angles at the centre are equal.
同一个圆中，相等的弦到圆心的距离相等，且所对的圆心角相等。逆命题同样成立：到圆心距离相等的弦相等，所对圆心角相等的弦也相等。
AB = CD ⇌ ∠AOB = ∠COD ⇌ equal arcs AB and CD
The same chain of equivalence extends to arcs: equal arcs correspond to equal chords and equal centre angles. This is why a regular polygon inscribed in a circle splits the 360° at the centre into equal parts of 360° ÷ n, and each chord is a side of the polygon.
这条等价链同样适用于弧：等弧对应等弦与等圆心角。正因如此，圆内接正多边形把圆心的 360° 均分为 360° ÷ n 份，每条弦就是多边形的一条边。
Note the important restriction: these results hold within one circle, or between circles of equal radius (congruent circles). They do not apply to two circles with different radii.
注意一个重要前提：这些结论只在同一个圆内，或半径相等的等圆之间成立，不适用于半径不同的两个圆。
Exam reason to write: "equal chords are equidistant from the centre" or "equal chords subtend equal angles at the centre".
考试中应写的理由："等弦到圆心等距"或"等弦所对圆心角相等"。
11. Writing Proofs That Score Full Marks | 证明书写与理由表述
Most circle theorem marks are awarded for the reason, not the number. A correct angle with no justification typically earns half credit; a correct angle with a precise reason earns full credit. Write the reason on the same line as the calculation, separated by a comma or the word "because".
圆定理的分数大多给在"理由"上，而不是数字上。角度算对但没写理由，通常只能拿一半分；角度正确且理由准确，才能拿满分。请把理由写在同一行，用逗号或"because"与算式隔开。
Learn the standard phrasing rather than improvising. Examiners accept short mark-scheme abbreviations, but they do not accept vague statements such as "circle rule" or "because of the picture". If you use the wrong theorem name with the right number you will usually still be penalised.
要背标准表述，不要临场自创。考官接受评分方案中的简短说法，但不接受"圆定理规则""因为图就是这样"这类含糊表述。用错定理名称即使数字正确，通常仍会被扣分。
Follow this four-step proof structure:
请遵循以下四步证明结构：
- Step 1 – Mark the given information on the diagram | 第一步：把已知信息标在图上
- Step 2 – State the theorem you are using by name | 第二步：写出你所用定理的名称
- Step 3 – Show the arithmetic or angle fact in a clear line | 第三步：用清晰的一行给出运算或角度关系
- Step 4 – Give the final answer with the correct unit ( ° for angles) | 第四步：给出最终答案并带正确单位（角度用 ° ）
A model line looks like this: "∠APB = 65°, angle at centre is twice angle at circumference." A weak line looks like this: "65°" The difference in marks between these two answers is routinely two or three per question.
规范的写法示例："∠APB = 65°，圆心角是圆周角的两倍。" 不合格的写法："65°"。这两者之间一题常常差两到三分。
Also remember to state the geometric fact before the conclusion in "show that" questions, and to use "since" and "therefore" correctly. When a question says "give a reason for each step", write a reason for every single line, even the obvious ones.
另外，在"证明题（show that）"中要先陈述几何事实再给结论，并正确使用"因为/所以"。当题目要求"每一步给出理由"时，每一行都要写理由，哪怕看起来很明显。
12. Common Mistakes and a Revision Checklist | 常见失分点与复习清单
The most frequent error is applying the centre-angle theorem to two angles that do not subtend the same arc. Before you double or halve anything, trace the arc with your finger and confirm that both angles open onto exactly that arc.
最常见的错误是把圆心角定理用在并非对同一段弧的两个角上。在乘 2 或除以 2 之前，请用手指沿着那段弧划一遍，确认两个角都对着同一段弧。
The second most frequent error is failing to spot the isosceles triangles formed by radii. Two radii drawn to any two points on a circle create an isosceles triangle, so the two base angles are equal. Examiners build this into diagrams constantly, and it is often the first step of a longer chain.
第二常见的错误是没有看出半径构成的等腰三角形。从圆心连向圆上任意两点，都会形成等腰三角形，两个底角相等。考官反复在图形中埋这个点，而它常常是整条推理链的第一步。
Other reliable marks-losers include: mixing up the diameter and the radius in a Pythagoras calculation; assuming a line passes through the centre when it is not marked as a diameter; ignoring the reflex angle case; rounding angles prematurely; and using the tangent-chord angle with the wrong segment.
其他稳定的失分点包括：在勾股计算中把直径与半径弄混；在图中未标明是直径时，擅自认为某条线过圆心；忽略优角的情形；过早对角度取近似；以及把弦切角用在了错误的弓形上。
Use this quick reference table to check yourself before the exam:
考前可用下表自查：
| Situation in the diagram | 图中的情形 | Theorem to use | 使用定理 | Result | 结论 |
|---|---|---|
| Centre angle and circumference angle on the same arc | 圆心角与圆周角对同弧 | Angle at the centre | 圆心角定理 | Centre angle = 2 × circumference angle |
| Triangle with one side as a diameter | 一边为直径的三角形 | Angle in a semicircle | 半圆圆周角 | The angle on the circumference is 90° |
| Two angles standing on the same chord, same side | 同弦同侧的两个角 | Same segment | 同弧圆周角 | style="border:1px
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