📚 Circle Theorems for IGCSE Maths | IGCSE 数学圆定理
Circle theorems are a core topic in the IGCSE Mathematics syllabus, including Cambridge 0580 and 0980 and Edexcel 4MA1. They describe the fixed angle relationships that exist between angles, chords, tangents, arcs and sectors both inside and around a circle. These rules appear in both calculator and non-calculator papers, most often as multi-step geometry problems, and in the extended papers as proof questions or "give a reason" questions where the mark depends entirely on your written justification. You therefore need two separate skills: knowing how to use each theorem to find a missing angle, and knowing how to state the correct geometric reason in the precise wording the mark scheme expects.
圆定理是 IGCSE 数学大纲中的核心内容，涵盖剑桥 0580 / 0980 以及爱德思 4MA1。它们描述了圆内部以及圆周围的角度、弦、切线、弧和扇形之间固定的角度关系。这些规则在计算器和非计算器试卷中都会出现，最常见的题型是多步几何题，而在拓展卷中还会以证明题或"说明理由"的形式出现，此时得分完全取决于你写出的论证过程。因此你需要掌握两项独立的技能：一是会运用每一条定理求出未知角，二是能用评分标准要求的准确措辞写出正确的几何理由。
1. Key Vocabulary and Circle Parts | 圆的关键术语与组成部分
Before applying any circle theorem, you must be able to identify and name the main parts of a circle correctly. The centre is the fixed point inside the circle from which all points on the circle are equidistant. The radius (r) is the distance from the centre to any point on the circumference. The diameter (d) is a special chord that passes through the centre, and it is exactly twice the radius.
在运用任何圆定理之前，你必须能够正确识别并说出圆的主要组成部分。圆心是圆内部的固定点，圆周上所有点到它的距离都相等。半径 (r) 是从圆心到圆周上任意一点的距离。直径 (d) 是一条经过圆心的特殊弦，长度恰好是半径的两倍。
d = 2r r = d ÷ 2
A chord is a line segment joining any two points on the circumference. A tangent is a straight line that touches the circle at exactly one point, called the point of contact, and never crosses into the interior of the circle. An arc is part of the circumference, a sector is the region enclosed by two radii and an arc, and a segment is the region enclosed by a chord and an arc.
弦是连接圆周上任意两点的线段。切线是一条与圆只有一个公共点的直线，这个公共点叫做切点，切线不会穿过圆的内部。弧是圆周的一部分，扇形是由两条半径和一段弧围成的区域，弓形（也叫弦切弓形）是由一条弦和一段弧围成的区域。
- Centre | 圆心
- Radius (r) | 半径 (r)
- Diameter (d = 2r) | 直径 (d = 2r)
- Chord | 弦
- Tangent | 切线
- Arc | 弧
- Sector | 扇形
- Segment | 弓形（弦与弧之间的区域）
- Point of contact | 切点
- Subtend | 对……张角 / 所对
- Cyclic quadrilateral | 圆内接四边形
- Concyclic points | 共圆点
Two further words appear in almost every circle theorem question. An angle subtends a chord or an arc when that chord or arc is the "base" the angle opens onto. A cyclic quadrilateral is a four-sided shape whose four vertices all lie on a single circle; the four vertices are then called concyclic points. Recognising these two ideas instantly tells you which theorem to reach for.
还有两个词几乎出现在每一道圆定理题目中。当一个角"张开"对着某条弦或某段弧时，我们说这个角对该弦或该弧张角。圆内接四边形是四个顶点都落在同一个圆上的四边形，这四个顶点称为共圆点。一旦能迅速识别这两种结构，你就能立刻判断该用哪条定理。
| Part | 组成 | Definition | 定义 |
| Centre 圆心 | The fixed point equidistant from every point on the circle. 到圆周各点距离相等的固定点。 |
| Radius 半径 | Distance from centre to circumference. 圆心到圆周的距离。 |
| Diameter 直径 | A chord through the centre; d = 2r. 过圆心的弦；d = 2r。 |
| Chord 弦 | Segment joining two points on the circumference. 连接圆周上两点的线段。 |
| Tangent 切线 | Line touching the circle at exactly one point. 只与圆有一个公共点的直线。 |
| Sector 扇形 | Region bounded by two radii and an arc. 两条半径与一段弧围成的区域。 |
| Segment 弓形 | Region bounded by a chord and an arc. 一条弦与一段弧围成的区域。 |
2. Angle in a Semicircle | 半圆所对的圆周角
The first and most heavily used theorem states that the angle in a semicircle is a right angle. If AB is a diameter of a circle and C is any other point on the circumference, then the angle at C is 90°. The reason this works is that the angle at the centre on the diameter is 180°, and the angle at the circumference is always half of it.
第一条也是使用最频繁的定理是：半圆所对的圆周角是直角。如果 AB 是圆的直径，C 是圆周上任意其他一点，那么 C 处的角等于 90°。原因是直径所对的圆心角是 180°，而圆周角总是它的一半。
∠ACB = 90°
In the exam you must write the reason, not just the number. The accepted wording is "angle in a semicircle is 90°" or "angle subtended by a diameter is a right angle". Writing only "90°" with no reason usually loses the method mark, especially in the extended paper.
考试中你必须写出理由，而不能只写数字。可接受的表述是"半圆上的圆周角是 90°"或"直径所对的圆周角是直角"。如果只写"90°"而不写理由，在拓展卷中通常会丢掉方法分。
The theorem also works in reverse, and this converse is examined regularly. If a triangle drawn with all three vertices on a circle has a right angle at one vertex, then the side opposite that right angle must be a diameter. This is the standard way to prove that a line passes through the centre of a circle.
该定理反过来也成立，而这个逆定理也是常考点。如果一个三角形的三个顶点都在圆上，且其中一个角是直角，那么直角所对的边一定是直径。这就是证明一条直线经过圆心的标准方法。
3. Angle at the Centre and Angle at the Circumference | 圆心角与圆周角
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 circumference. This single theorem underpins several of the others, so it is worth understanding rather than memorising.
同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。这一条定理是其他几条定理的基础，因此值得理解而不是死记。
∠AOB = 2 × ∠ACB
Be careful about which angle at the centre you use. The "angle at the centre" must be the one that subtends the same arc as the angle at the circumference. If the given central angle is the reflex angle (greater than 180°), then the non-reflex angle is 360° minus that value, and you should use the non-reflex one when the point C lies on the minor arc.
要特别注意该用哪个圆心角。所谓"圆心角"必须是与圆周角对同一段弧的那个角。如果题目给出的圆心角是大于 180° 的优角，那么另一个圆心角等于 360° 减去它；当点 C 位于劣弧上时，应使用这个非优角。
Worked example: if ∠AOB = 130°, then ∠ACB = 130° ÷ 2 = 65° when C is on the major arc. If ∠OB is given as the reflex angle 260°, the relevant central angle is 360° − 260° = 100°, so ∠ACB = 50°.
例题：若 ∠AOB = 130°，且 C 在优弧上，则 ∠ACB = 130° ÷ 2 = 65°。若给出的 ∠AOB 是优角 260°，则相关圆心角为 360° − 260° = 100°，于是 ∠ACB = 50°。
4. Angles in the Same Segment | 同弧所对的圆周角相等
Angles in the same segment are equal. If two points C and D lie on the same arc on the same side of a chord AB, then the angles ∠ACB and ∠ADB are equal, because both are half of the same angle at the centre.
同一弓形内的圆周角相等。如果两点 C 和 D 位于弦 AB 同一侧的同一条弧上，那么 ∠ACB 与 ∠ADB 相等，因为它们都是同一个圆心角的一半。
∠ACB = ∠ADB
The official reason to write is "angles in the same segment are equal" or "angles subtended by the same arc are equal". A very common trap is to apply this theorem when the two angles actually lie on opposite sides of the chord. In that case the two angles are not equal; they are supplementary, because they form a cyclic quadrilateral.
标准的理由表述是"同弓形内的圆周角相等"或"同弧所对的圆周角相等"。一个非常常见的陷阱是：当两个角实际位于弦的异侧时仍套用此定理。此时两角并不相等，而是互补，因为它们构成一个圆内接四边形。
This theorem is also the fastest route to proving that four points are concyclic. If two points on the same side of a line segment subtend equal angles at that segment, then all four points lie on one circle.
这条定理也是证明四点共圆的最快捷方法。如果两点在线段的同一侧对该线段所张的角相等，那么这四点共圆。
5. Cyclic Quadrilaterals: Opposite Angles | 圆内接四边形对角互补
A cyclic quadrilateral is any four-sided shape with all four vertices on a circle. Its key property is that each pair of opposite angles adds up to 180°. This follows directly from the double-angle theorem: opposite angles in the quadrilateral stand on complementary arcs, and the two central angles involved add up to 360°.
圆内接四边形是四个顶点都在同一个圆上的四边形。它的关键性质是每一组对角之和都等于 180°。这直接来自圆心角定理：四边形的对角所对的弧互补，而对应的两个圆心角之和为 360°。
∠A + ∠C = 180° ∠B + ∠D = 180°
Typical exam wording is "opposite angles of a cyclic quadrilateral sum to 180°". If you are told only that a shape "has all four vertices on the circle", you are entitled to use this rule immediately. If the shape is not described as cyclic, you cannot assume the property holds.
考题中标准的理由表述是"圆内接四边形对角互补"。如果题目只告诉你某个四边形的四个顶点都在圆上，你就可以立即使用这条规则。但如果图形没有被说明是圆内接四边形，就不能假定该性质成立。
Remember that a rectangle and an isosceles trapezium drawn with a circle through all four vertices are automatically cyclic, since each has opposite angles of 90° or equal base angles that satisfy the 180° condition.
请记住：如果矩形或等腰梯形能作出过四个顶点的圆，它们自动是圆内接四边形，因为它们要么对角都是 90°，要么底角相等，都满足 180° 的条件。
6. Cyclic Quadrilaterals: The Exterior Angle | 圆内接四边形的外角
There is a second, equally examinable property of cyclic quadrilaterals. If one side of a cyclic quadrilateral is extended, the exterior angle formed is equal to the interior opposite angle of the quadrilateral.
圆内接四边形还有第二条同样常考的性质。如果延长圆内接四边形的一条边，所形成的外角等于四边形的内对角。
∠DCE = ∠A
The reason is short and reliable: the exterior angle and its adjacent interior angle are supplementary, and the interior opposite angle is also supplementary to that same interior angle. Therefore the two are equal. In the exam, write "exterior angle of a cyclic quadrilateral equals the interior opposite angle".
理由简短而可靠：外角与其相邻内角互补，而内对角也与同一个内角互补，因此两者相等。考试中应写出"圆内接四边形的外角等于内对角"。
Many students lose marks here simply by using the 180° version when the question gives an extended side. Read whether the angle is inside the quadrilateral or outside it before choosing the theorem.
许多学生在这里丢分，仅仅是因为题目延长了一边时仍使用 180° 那一条。在看图时先判断这个角是在四边形内部还是在外部，再选择对应的定理。
7. Tangent Properties | 切线的性质
A tangent meets a circle at exactly one point, and the most important consequence is that the tangent is perpendicular to the radius drawn to the point of contact. Whenever you see a tangent and a radius meeting at a point on the circle, mark a right angle there immediately.
切线与圆只有一个公共点，最重要的推论是：切线垂直于过切点的半径。只要看到切线与半径在圆上某点相交，就应立刻在该处标出直角。
∠OTA = 90°
If two tangents are drawn from the same external point to a circle, three facts follow at once. The two tangent lengths are equal; the line from the external point to the centre bisects the angle between the two tangents; and it also bisects the angle between the two radii at the centre. These three facts together with the right angle usually give you two congruent right-angled triangles, which is why examiners like this configuration.
如果从同一个圆外一点向圆引两条切线，可以立刻得到三个结论：两条切线长度相等；圆外点与圆心的连线平分两条切线的夹角；这条连线也平分圆心处两条半径的夹角。这三个结论加上直角，通常能给出两个全等的直角三角形，这也是考官偏爱这种图形的原因。
- Tangent ⟂ radius at the point of contact | 切线垂直于过切点的半径
- Two tangents from an external point are equal in length | 圆外一点引出两条切线长度相等
- The line to the centre bisects the angle between the tangents | 与圆心的连线平分两切线夹角
- Two congruent right-angled triangles are formed | 形成两个全等的直角三角形
A useful combination question is a tangent together with the alternate segment theorem, or a tangent together with the angle-in-a-semicircle theorem. In both cases the right angle at the point of contact is the starting point of the whole solution.
常见的组合题是把切线与弦切角定理结合，或者与"半圆上的圆周角"结合。在这两种情形中，切点处的直角都是整个解题过程的起点。
8. The Alternate Segment Theorem | 弦切角定理（交替弓形定理）
The alternate segment theorem states that the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. In other words, the angle between the tangent and chord AB equals the angle that any point on the opposite arc of chord AB makes with AB.
弦切角定理指出：切线与弦在切点处所夹的角，等于另一弓形内的圆周角。换言之，切线与弦 AB 的夹角，等于弦 AB 另一侧弧上任意一点对 AB 所张的角。
∠(tangent, AB) = ∠ACB
The word "alternate" is essential. There are two segments created by the chord AB, and the angle you compare with must lie in the segment on the other side of the chord from the tangent. Mixing up the two segments is the single most common mistake with this theorem.
"另一弓形"这个词至关重要。弦 AB 把圆分成两个弓形，你所比较的角必须位于切线的另一侧弓形中。把两个弓形弄混是这条定理最常见的错误。
The reason to write is "alternate segment theorem" or "angle between tangent and chord equals angle in the alternate segment". This theorem is almost always combined with the isosceles triangle formed by two radii, so be ready to use the fact that base angles of an isosceles triangle are equal.
应写出的理由是"弦切角定理"或"切线与弦的夹角等于另一弓形内的圆周角"。这条定理几乎总是与两条半径构成的等腰三角形一起使用，所以要随时准备用"等腰三角形两底角相等"这一性质。
9. Chord Properties and the Centre | 弦的性质与圆心
A perpendicular drawn from the centre of a circle to a chord bisects that chord. The converse is equally useful: the perpendicular bisector of any chord passes through the centre of the circle, which is the standard construction for finding the centre.
从圆心向弦作垂线，必平分这条弦。逆定理同样有用：任意一条弦的垂直平分线必经过圆心，这也是找圆心的标准作图方法。
Two related facts appear in exam questions. First, equal chords are equidistant from the centre, and chords that are equidistant from the centre are equal. Second, equal chords subtend equal angles at the centre, and conversely equal angles at the centre are subtended by equal chords.
考试中还会出现两个相关结论。第一，等弦到圆心的距离相等，反过来，到圆心距离相等的弦相等。第二，等弦所对的圆心角相等，反过来，相等的圆心角所对的弦相等。
- Perpendicular from centre bisects the chord | 圆心到弦的垂线平分该弦
- Perpendicular bisector of a chord passes through the centre | 弦的垂直平分线过圆心
- Equal chords are equidistant from the centre | 等弦到圆心距离相等
- Equal chords subtend equal angles at the centre | 等弦所对的圆心角相等
When a question gives a chord length and a distance from the centre, you are almost certainly expected to form a right-angled triangle using half the chord, the distance, and the radius, and then apply Pythagoras. The radius is always the hypotenuse.
当题目给出弦长和弦到圆心的距离时，几乎可以肯定你需要用半条弦、该距离和半径构成直角三角形，然后运用勾股定理。其中半径永远是斜边。
10. Extension: Intersecting Chords and Power of a Point | 拓展：相交弦定理与点幂
Strictly speaking, intersecting chords and secant-tangent relationships lie beyond the core IGCSE circle theorem list, but they appear in IGCSE Additional Mathematics, in some Further Mathematics specifications, and at the start of A Level. Knowing them gives you a real advantage and makes the standard theorems easier to see as one connected family.
严格来说，相交弦定理与割线—切线关系超出了 IGCSE 核心圆定理的范围，但它们出现在 IGCSE 附加数学、部分进阶数学大纲以及 A Level 的开头部分。掌握它们能带来真正的优势，也能让你看出标准定理其实属于同一个相互联系的整体。
If two chords AB and CD intersect at a point X inside the circle, the products of the two parts of each chord are equal.
如果两条弦 AB 与 CD 在圆内一点 X 相交，那么每条弦被分成的两段之积相等。
AX × XB = CX × XD
If instead a tangent from an external point P touches the circle at T, and a secant from P cuts the circle at B and C, then the square of the tangent length equals the product of the two secant segments.
如果从圆外一点 P 引出的切线在
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