📚 Shape and Space 10: Circle Theorems | 形状与空间 10：圆定理
Welcome to Shape and Space 10. In this unit, we take a close look at circle theorems – a core part of Edexcel IGCSE Mathematics. Circle theorems describe special angle and segment properties within circles and are essential for solving geometric problems without the need for measurement. Understanding these rules will help you find missing angles, prove geometric relationships, and tackle multi-step exam questions with confidence. Let's dive into the fundamental theorems and see how they connect.
欢迎学习形状与空间10。本单元我们深入探讨圆定理——Edexcel IGCSE数学的核心内容之一。圆定理描述了圆内部角度和线段的特殊性质，是无需测量即可解决几何问题的关键。掌握这些规律，你就能熟练求解未知角度、证明几何关系，以及自信应对多步推理的考试题。让我们一起学习这些基本定理，并理清它们之间的关联。
1. Angle at the Centre Theorem | 圆心角定理
umference. To apply the theorem correctly, you must be certain that both angles are standing on the same arc. In diagrams, the central angle is formed by two radii, while the angle at the circumference has its vertex on the circle.
同一弧所对的圆心角是圆周角的2倍。我们通常记作：圆心角 = 2 × 圆周角。正确应用这条定理的前提是确认两个角对的是同一条弧。在图中，圆心角由两条半径构成，而圆周角的顶点落在圆周上。
Angle at centre = 2 x Angle at circumference
圆心角 = 2 × 圆周角
Be careful: If the arc is changed, the relationship no longer holds. Also, when the angle at the centre is given as a reflex angle (greater than 180 degree), we usually work with the corresponding acute or obtuse inscribed angle that stands on the minor arc.
注意：一旦所对的弧不同，这个倍数关系就不成立了。另外，如果给出的圆心角是优角（大于180°），我们通常使用劣弧所对的较小的圆周角来建立方程。
2. Angle in a Semicircle | 半圆上的圆周角
When a triangle is inscribed in a semicircle and its hypotenuse is the diameter, the angle opposite the diameter is always a right angle (90 degree). This is a special case of the angle at the centre theorem: the diameter produces a central angle of 180 degree, so the inscribed angle is 90 degree. This theorem is frequently used to prove that a line is a diameter or to spot right angles in circle diagrams.
若三角形内接于半圆，且斜边为直径，则直径所对的圆周角必定是直角（90°）。这是圆心角定理的特例：直径所对的圆心角为180°，因此圆周角为90°。该定理常用于证明某条弦是直径，或在圆的图形中识别直角。
90 degree. This works in both directions: if you are told that an inscribed angle equals 90 degree, then the chord opposite it must be a diameter.
对于圆周上任一点C，若AB是直径，则∠ACB = 90°。反之也成立：如果已知某圆周角为90°，则它所对的弦一定是直径。
3. Angles in the Same Segment | 同弧上的圆周角
Angles in the same segment of a circle are equal. More precisely, angles subtended by the same chord (or arc) and lying on the same side of the chord are equal. This theorem is extremely powerful when you need to transfer an angle value from one part of the diagram to another. If several angles are drawn on the same arc, all of them share the same measure.
同弧（或同弦）所对的圆周角相等。更准确地说，由同一条弦所对、且位于弦同一侧的所有圆周角都相等。当需要将某个角的值从图形一处转移到另一处时，这个定理非常有用。如果几个角都立在相同的弧上，它们的大小一定相同。
Look for a 'bow-tie' shape or a pair of angles with their vertices on the circle, both open to the same chord. This configuration usually signals that the equal-angle theorem applies.
寻找'蝴蝶形'或两个顶点都在圆上、开口朝着同一条弦的角。这种结构通常意味着相等圆周角定理能够派上用场。
4. Cyclic Quadrilateral | 圆内接四边形
A quadrilateral whose four vertices all lie on a circle is called a cyclic quadrilateral. The key properties are: opposite angles sum to 180 degree (they are supplementary), and an exterior angle equals the interior opposite angle. These facts allow you to set up equations linking unknown angles and to prove that four points are concyclic.
四个顶点都在同一个圆上的四边形叫做圆内接四边形。其核心性质是：对角互补（和为180°），以及一个外角等于它的内对角。利用这些性质可以建立未知角的方程，还能证明四点共圆。
ditionally, if we extend side AB to E, then angle CBE = angle D (interior opposite angle).
对于圆内接四边形ABCD，∠A + ∠C = 180°，∠B + ∠D = 180°。此外，如果延长边AB到E，那么外角∠CBE等于内对角∠D。
5. Tangent and Radius | 切线与半径
ngle OTP = 90 degree. This right angle is often the starting point for applying Pythagoras' theorem or trigonometric ratios in circle problems.
圆的切线是一条与圆只有一个公共点的直线。切线与半径定理告诉我们，连接圆心和切点的半径垂直于切线。因此，如果OT是半径，TP是切线，切点为T，那么∠OTP = 90°。这个直角常常是运用勾股定理或三角比解决圆的问题的突破口。
Remember: a tangent can only touch a circle at one point. Any line that cuts through a circle is a secant, not a tangent. The right angle condition is crucial for constructing and reasoning about distances from an external point.
请记住：切线仅在一点接触圆。任何穿过圆的直线都是割线，不是切线。切线与半径的垂直关系对于从外部点构建图形和推理距离至关重要。
6. Tangents from an External Point | 从圆外一点引切线
gle UPO. This symmetry often helps in solving problems involving perimeter or angle chase.
从圆外一点P可以引圆的两条切线，切点分别为T和U。线段PT和PU的长度相等。此外，连接P和圆心O的直线平分两切线之间的夹角，即∠TPO = ∠UPO。这种对称性经常用于解决涉及周长或角度推算的问题。
If you are asked to find the length of a tangent segment, draw the radii OT and OU. Then triangles OTP and OUP are congruent right-angled triangles, allowing you to use Pythagoras' theorem or trigonometry with the distance OP.
如果要求切线段的长度，可以画出半径OT和OU。由此构成的三角形OTP和OUP是全等的直角三角形，这样你就可以利用勾股定理或三角函数结合OP的长度进行求解。
7. Alternate Segment Theorem | 弦切角定理
The alternate segment theorem is one of the most powerful and frequently examined circle theorems. It states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment – that is, the angle made by the chord in the opposite arc. For example, if AT is the tangent at point A, and AB is a chord, then angle TAB equals the angle in the circle on the other side of AB, say angle ACB.
弦切角定理是最强大、最常见的考试定理之一。它指出：切线与过切点的弦所夹的角等于弦在另一侧弧所对的圆周角（即'弦切角等于内对角'）。例如，若AT是切于点A的切线，AB是弦，则∠TAB等于弦AB另一侧所对的圆周角，比如∠ACB。
This theorem is particularly useful when the diagram includes a tangent and a chord, and you need to relate an angle outside the circle to one inside. Always check that the angle you are comparing sits on the opposite arc, not the same side.
当图形中同时出现切线和弦，且需要将圆外的角与圆内的角联系起来时，这个定理尤其有用。务必确认所比较的角位于弦的另一侧弧上，而不是同侧。
8. Intersecting Chords Theorem | 相交弦定理
mes from the similarity of triangles formed by the intersecting chords. It is not limited to diameters or special chords; any pair of intersecting chords obeys this rule.
如果两条弦AB和CD在圆内相交于点X，那么每条弦被交点分成的两段长度的乘积相等：AX · XB = CX · XD。这个定理源于相交弦所构成的相似三角形。它不限于直径或特殊弦；任何一对相交弦都满足这个规律。
In IGCSE, problems often supply three segment lengths and ask for the fourth. Simply set up the equation and solve. Also, if one chord passes through the centre, the theorem still applies, but be careful to use the full segment lengths, not just the radius.
在IGCSE考试中，题目通常给出三条线段长，要求求出第四条。只需列出方程求解即可。此外，如果一条弦经过圆心，该定理同样适用，但要注意使用完整的弦段长度，而非仅仅半径的长度。
9. Using Algebra with Circle Theorems | 代数与圆定理的结合
Solving yields x, and then the actual angles can be found. The same approach works with the angle at centre theorem, equal angles in the same segment, or any relationship that links angles.
许多IGCSE题目会将圆定理与代数结合起来。你可能会看到用含有x的表达式，比如（2x + 10）°和（4x - 20）°代表圆内接四边形的一组对角。利用对角互补，可以列出方程：(2x + 10) + (4x - 20) = 180。解出x后，就能算出各个角的实际度数。同样的方法也适用于圆心角定理、同弧上圆周角相等等任何角度关系。
Always state which theorem you are using before setting up the equation. This shows clear reasoning and helps avoid mistakes. Also, remember to check whether the angles belong to the correct configuration – for instance, opposite angles must be used for cyclic quadrilaterals, not adjacent ones.
在列方程前务必先说明你使用的是哪一条定理。这样既能展现清晰的推理过程，又能避免错误。此外，要反复确认所选的角是否属于正确的配置——例如，圆内接四边形需要使用对角，而不是邻角。
10. Worked Example | 典型例题
Problem: B and chord CD intersect at X. AX = 3 cm, XB = 8 cm, and CX = 4 cm. Find the length of XD.
题目：在圆中，弦AB与弦CD交于点X。已知AX = 3 cm, XB = 8 cm, CX = 4 cm。求XD的长度。
Solution: ting the values: 3 x 8 = 4 x XD, so 24 = 4 x XD. Dividing both sides by 4 gives XD = 6 cm.
解：根据相交弦定理，AX · XB = CX · XD。代入数值：3 × 8 = 4 × XD，得24 = 4 × XD。两边同除以4，得XD = 6 cm。
Problem 2: Points A, B, C and D lie on a circle. Tangent at A meets chord AC at angle 35 degree. Find angle ABC, given that C is on the major arc.
问题2：点A、B、C、D共圆。过点A的切线与弦AC的夹角为35°。若点C在优弧上，求∠ABC。
Solution: ABC in the alternate segment. So angle ABC = 35 degree. No further calculation is needed.
解：切线AT与弦AC的夹角为35°。由弦切角定理，这个角等于另一侧弧所对的圆周角∠ABC。因此∠ABC = 35°。无需更多计算。
11. Summary of Circle Theorems | 圆定理汇总
The table below summarises the key theorems you must memorise for the Edexcel IGCSE Shape and Space section. Use it as a quick reference while revising.
下表汇总了Edexcel IGCSE形状与空间部分必须记忆的核心定理，复习时可作为快速参考。
| Theorem 定理 | Statement 内容 |
|---|---|
| Angle at centre | 圆心角 = 2 × 圆周角 (same arc) |
| Angle in semicircle | 直径所对的圆周角是90° |
| Angles in same segment | 同弧上圆周角相等 |
| Cyclic quadrilateral | 对角互补，外角=内对角 |
| Tangent-radius | 切线垂直于过切点的半径 |
| Tangents from a point | 切线长相等；平分夹角 |
| Alternate segment | 弦切角 = 内对角（另侧圆周角） |
| Intersecting chords | AX · XB = CX · XD (弦段乘积相等) |
Memorising the exact wording is less important than being able to recognise the configuration and apply the theorem correctly. Always draw a sketch if the question does not provide a diagram, and label angles clearly.
比起死记硬背文字，更重要的是能识别定理的图形特征并正确应用。如果题目没有给出图，一定自己画草图并清楚地标出角度。
12. Exam Tips and Common Pitfalls | 应试技巧与常见错误
1. Identify the correct arc or chord first. Many mistakes happen because students apply a theorem to the wrong pair of angles. Always trace the arc or chord that subtends the angles you are comparing.
1. 先确定正确的弧或弦。很多错误是由于学生将定理错误地应用于不对应的角。务必追踪所比较的角到底是对哪一条弧或弦。
2. Look for right angles. Check for tangents touching radii, diameters, or semi-circles. Right angles unlock Pythagoras and trigonometry, and they often form the bridge between circle theorems and other geometry topics.
2. 寻找直角。留意切线与半径接触、直径、半圆等结构。直角能让你运用勾股定理和三角学，它们常常是连接圆定理与其他几何内容的桥梁。
3. Beware of 'not drawn accurately' diagrams. In Edexcel IGCSE, diagrams may be illustrative but not precise. Do not guess an angle measure from the drawing; always calculate using theorems.
3. 当心'未精确绘制'的图形。在Edexcel IGCSE考试中，图示可能只是示意，并非精确。不要凭图猜测角度大小，一定要用定理计算。
4. State the theorem. When answering 'show that' or 'prove' questions, a short justification such as 'angle at centre = 2x angle at circumference' od earns method marks. Even in calculation questions, jotting down the theorem used can prevent errors.
4. 写出定理名称或内容。在回答'证明'类题目时，简单写一句'圆心角=2×圆周角'就能拿到方法分。就算在计算题中，简单注明所用定理也能避免失误。 Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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