📚 Mastering Circle Theorems: IGCSE G-4 Teacher Notes | IGCSE 圆定理精讲：G-4 教师笔记
Circle theorems are one of the most visual and logical parts of IGCSE Mathematics. They describe angle and length relationships within circles, and they appear regularly on both Core and Extended papers. This revision article is designed for teachers and students following the IGCSE syllabus unit G-4. It explains the key theorems, shows how to recognise them in diagrams, and builds the problem-solving skills needed for high-scoring answers.
圆定理是 IGCSE 数学中最直观、最讲逻辑的内容之一。它们描述圆内角度和线段之间的关系，经常出现在核心和拓展试卷中。这篇复习文章面向 IGCSE 大纲 G-4 单元的教师和学生，讲解关键定理，展示如何在图形中识别它们，并培养获得高分所需的解题能力。
Each section below pairs the mathematical statement with the exact wording an examiner expects, and every theorem is followed by the diagram cue that tells you it is the right tool to use. Teachers can read the sections in order as a lesson sequence; students can jump straight to the recognition checklist and the worked example when revising the night before the exam.
下文每一节都把数学结论与考官期待看到的准确表述配对呈现，每条定理后面都附有'图形线索'，告诉你何时该用它。教师可以按顺序把各节当作一个教学序列来读；学生考前一晚复习时，可以直接跳到识别清单和例题部分。
1. What Are Circle Theorems? | 什么是圆定理？
Circle theorems are rules about angles and lengths formed by lines such as chords, tangents, and radii inside or around a circle. Problems usually ask you to find an unknown angle marked x or y. To solve them, you must first identify which theorem applies, then write the appropriate equation or reason.
圆定理是关于圆内或圆上由弦、切线和半径等线段形成的角与长度的规则。题目通常要求你求出标有 x 或 y 的未知角。解题时，你必须先识别适用哪条定理，然后写出相应的方程或理由。
Theorems are accepted as shortcuts: once you know the rule, you do not need to prove it every time in an IGCSE exam. However, you must quote the correct reason, such as 'angles in the same segment are equal'. Marks are split between the numerical answer and the stated reason, so a bare number without justification often loses half the credit.
这些定理被当作捷径使用：在 IGCSE 考试中，一旦你掌握规则，就不需要每次证明它。但你必须写出正确的理由，例如'同弓形内的圆周角相等'。分数被分配给数值答案和所写理由两部分，因此只写一个数字而没有依据，常常会丢掉一半的分。
In IGCSE unit G-4, the most commonly examined theorems include the angle at the centre, angles in the same segment, the angle in a semicircle, cyclic quadrilaterals, tangent-radius properties, two tangents from an external point, and the alternate segment theorem. The full list is set out below and every one of them is developed in its own section.
在 IGCSE 大纲 G-4 单元中，最常考的定理包括：圆心角定理、同弓形圆周角定理、半圆上的圆周角定理、圆内接四边形定理、切线与半径的性质、外部一点引出的两条切线，以及交切角（交替弓形）定理。完整清单列在下面，每条定理都会在单独的一节中展开讲解。
- The angle at the centre is twice the angle at the circumference subtended by the same arc.
圆心角等于同一段弧所对圆周角的两倍。 - Angles in the same segment are equal.
同一弓形内的圆周角相等。 - The angle in a semicircle is 90°.
半圆上的圆周角是 90°。 - Opposite angles of a cyclic quadrilateral add up to 180°.
圆内接四边形的对角之和为 180°。 - A tangent is perpendicular to the radius at the point of contact.
切线与过切点的半径垂直。 - Two tangents drawn from an external point are equal in length.
从圆外一点引出的两条切线长度相等。 - The angle between a tangent and a chord equals the angle in the alternate segment.
切线与弦的夹角等于另一弓形内的圆周角。
2. The Vocabulary You Need Before the Theorems | 学定理前必须掌握的术语
Most lost marks in G-4 questions come from misreading the diagram rather than from the algebra. A chord is a straight line joining two points on the circumference; a tangent touches the circle at exactly one point, called the point of contact. The angle subtended by an arc at a point means the angle formed at that point by the two chords drawn to the ends of the arc.
G-4 题目中丢分最多的情况，往往不是代数出错，而是读错了图形。弦是连接圆上两点的直线段；切线只与圆有一个公共点，该点称为切点。某段弧在一点所对的角，指的是从该点向弧的两个端点各引一条弦，这两条弦形成的角。
The circle is also divided into named regions. A segment is the area between a chord and an arc, and it is described as major or minor depending on its size; two angles standing on the same chord but in different segments are said to be in alternate segments. A sector is the 'pizza slice' between two radii and an arc, and a cyclic quadrilateral is a four-sided shape whose four vertices all lie on the circle - the vertices are described as concyclic.
圆还被划分为若干有名字的区域。弓形是弦与弧之间的区域，按大小分为优弓形和劣弓形；站在同一条弦上但位于不同弓形内的两个角，称为交替弓形内的角。扇形是两条半径与一段弧之间的'披萨块'区域；圆内接四边形是四个顶点都在圆上的四边形，这些顶点称为共圆点。
- Radius: a line from the centre to the circumference; all radii of one circle are equal.
半径：从圆心到圆周的线段；同一个圆的所有半径都相等。 - Diameter: a chord passing through the centre, equal to twice the radius.
直径：经过圆心的弦，长度等于半径的两倍。 - Chord: a straight line with both ends on the circumference; the longest chord is the diameter.
弦：两端都在圆周上的直线段；最长的弦是直径。 - Tangent: a line that touches the circle at exactly one point, the point of contact.
切线：与圆只有一个公共点的直线，该公共点叫切点。 - Subtend: an arc or chord subtends an angle at a point when that point is joined to both ends.
对：当某点与一段弧或一条弦的两端相连时，就说这段弧或弦在该点对角。 - Cyclic quadrilateral: a quadrilateral with all four vertices on the same circle.
圆内接四边形：四个顶点都在同一个圆上的四边形。
3. Theorem 1: The Angle at the Centre | 定理一：圆心角定理
If two points A and B lie on a circle with centre O, and C is any other point on the major arc, then the angle at the centre is exactly twice the angle at the circumference standing on the same arc. This is the parent theorem: several others, including the semicircle and cyclic quadrilateral results, are special cases of it.
若 A、B 是圆心为 O 的圆上的两点，C 是优弧上任意一点，则圆心角恰好等于同一段弧所对圆周角的两倍。这是'母定理'：半圆定理和圆内接四边形定理等，都是它的特例。
∠AOB = 2 × ∠ACB
The reason you should write is 'the angle at the centre is twice the angle at the circumference subtended by the same arc'. Remember the reflex case: if the angle at the centre you need is the one greater than 180°, it is still twice the angle at the circumference, so the two centre angles for the same chord add up to 360°.
你应当写下的理由是'圆心角等于同一段弧所对圆周角的两倍'。注意优角情形：若需要的圆心角大于 180°，它同样等于圆周角的两倍，因此同一条弦对应的两个圆心角之和为 360°。
The proof is short enough to quote in a' show that' Triangle AOC is isosceles because OA = OC, both being radii, so its base angles are equal: ∠OAC = ∠OCA. The exterior angle ∠AOD of triangle AOC equals the sum of the two interior opposite angles, so ∠AOD = 2 × ∠OCA. Repeating the argument for triangle BOC gives ∠BOD = 2 × ∠OCB, and adding the two results produces ∠AOB = 2 × ∠ACB.
这条定理的证明很短，可以在'证明'类题目中引用。连接 CO 并延长到点 D。因为 OA = OC（都是半径），三角形 AOC 是等腰三角形，所以两底角相等：∠OAC = ∠OCA。三角形 AOC 的外角 ∠AOD 等于两个不相邻内角之和，故 ∠AOD = 2 × ∠OCA。对三角形 BOC 重复同样推理得 ∠BOD = 2 × ∠OCB，两式相加即得 ∠AOB = 2 × ∠ACB。
4. Theorem 2: Angles in the Same Segment | 定理二：同弓形内的圆周角
Angles subtended by the same chord and lying in the same segment are equal. In the diagram this usually appears as two or more angles 'hanging' from the same chord and both opening towards the same side of the circle, often drawn with arrowheads to show which angles are being compared.
同一条弦所对、且位于同一弓形内的圆周角相等。在图形中，这通常表现为两个或多个角'挂'在同一条弦上，并且都朝向圆的同一侧，图中常用箭头标出要比较的角。
∠ACB = ∠ADB (C and D in the same segment)
Write the reason as 'angles in the same segment are equal'. Teachers should stress the phrase the same segment: if the two angles are in opposite segments, they are not equal but supplementary, because their four end points form a cyclic quadrilateral.
理由要写成'同弓形内的圆周角相等'。教师应强调'同一弓形'这个限定：如果两个角位于相对的弓形内，它们并不相等，而是互补，因为它们的四个端点构成一个圆内接四边形。
The quickest way to see this theorem is to apply Theorem 1 twice. If O is the centre, then ∠ACB = ½ × ∠AOB and ∠ADB = ½ × ∠ class: students who understand that most circle theorems are chained together rarely confuse which reason to quote.
理解这条定理最快的方法是把定理一用两次。若 O 为圆心，则 ∠ACB = ½ × ∠AOB，∠ADB = ½ × ∠AOB，因为两角都对着同一段弧 AB。既然两者都等于同一个圆心角的一半，它们必然相等。这个联系值得在课堂上明说：明白'多数圆定理彼此串联'的学生，很少会混淆该写哪条理由。
5. Theorem 3: The Angle in a Semicircle | 定理三：半圆上的圆周角
If AB is a diameter and C is any point on the circle other than A or B, then the angle at C is a right angle. This is the case of Theorem 1 in which the angle at the centre is a straight angle of 180°, so the angle at the circumference is half of that, namely 90°.
若 AB 是直径，C 是圆上异于 A、B 的任意一点，则 C 处的角为直角。这其实是定理一在圆心角为 180° 平角时的特例，因此圆周角是它的一半，即 90°。
∠ACB = 90° when AB is a diameter
The converse is equally examinable: if a triangle inscribed in a circle has a right angle, then the side opposite that right angle is a diameter, and its midpoint is the centre. The reason to write is 'the angle in a semicircle is 90°' or, for the converse, 'the angle in a semicircle is a right angle, so AB is a diameter'.
逆定理同样常考：若内接于圆的三角形有一个直角，则这个直角所对的边就是直径，其中点即圆心。理由写'the angle in a semicircle is 90°'；若是逆用，则写'the angle in a semicircle is a right angle, so AB is a diameter'。
Because the triangle ACB is right-angled at C, Pythagoras' CB². This is the bridge between G-4 circle theorems and coordinate geometry questions, where you may be told three points and asked to prove that the angle at one of them is a right angle. A useful numerical habit is to check 3-4-5 or 5-12-13 triangles, since examiners like to hide a familiar triple inside a circle.
由于三角形 ACB 在 C 处为直角，勾股定理在它内部成立：AB² = AC² + CB²。这就是 G-4 圆定理与坐标几何题之间的桥梁——题目可能给出三个点，要求证明其中一点处的角是直角。一个实用的数值习惯是检查 3-4-5 或 5-12-13 三角形，因为考官喜欢把常见的勾股数藏在圆里。
6. Theorem 4: Cyclic Quadrilaterals | 定理四：圆内接四边形
When all four vertices of a quadrilateral lie on a circle, the two pairs of opposite angles are supplementary, that is, each pair adds to 180°. A second, closely related fact is that the exterior angle at a vertex equals the interior opposite angle, which is simply the first fact disguised by angles on a straight line.
当四边形的四个顶点都在同一个圆上时，两组对角分别互补，即每一组对角之和为 180°。与它密切相关的第二个事实是：某一顶点处的外角等于其内对角，这其实就是第一条事实用'平角'伪装后的版本。
∠A + ∠C = 180° and ∠B + ∠D = 180°
The standard reason is 'opposite angles of a cyclic quadrilateral add up to 180°'. For the exterior version write 'the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle'. Both phrases must be learned verbatim, because examiners accept the meaning but students who write 'it is a cyclic quadrilateral, so the angles are 180°' frequently lose the communication mark.
标准理由是'圆内接四边形的对角之和为 180°'。若是外角形式，则写'圆内接四边形的外角等于它的内对角'。这两句话都必须逐字记牢，因为考官虽然接受其含义，但学生若写成'它是圆内接四边形，所以角是 180°'，往往会丢掉表达分。
. Join the centre O to two opposite vertices, say A and C. The angle at the circumference, ∠D, is half of the reflex angle ∠AOC, while ∠B is half of the non-reflex ∠ . The converse is also a valid test: if one pair of opposite angles of a quadrilateral adds to 180°, then the four points are concyclic.
证明是定理一的一次漂亮应用。连接圆心 O 与一组对角顶点，例如 A 和 C。圆周角 ∠D 等于优角 ∠AOC 的一半，而 ∠B 等于非优角 ∠AOC 的一半。两个一半相加即为整个周角的一半，也就是 ½ × 360° = 180°。逆命题同样是有效的判定方法：若四边形有一组对角之和为 180°，则这四个点共圆。
7. Theorems 5 and 6: Tangents and Radii | 定理五与定理六：切线与半径
A tangent meets a
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