📚 Circle Theorems for IGCSE WJEC Maths | IGCSE WJEC 数学：圆定理考点精讲
Circle theorems are a set of rules that describe relationships between angles, chords, tangents, and arcs within a circle. In the IGCSE WJEC Mathematics specification, you are expected to know and apply these theorems to calculate missing angles and prove geometric statements. Mastering them not only helps you secure marks in the geometry section but also strengthens your logical reasoning for multi‑step problems.
圆定理是描述圆内角、弦、切线和弧之间关系的一组规则。在 IGCSE WJEC 数学考试中，你需要熟记并应用这些定理来计算未知角度、证明几何结论。掌握它们不仅能帮你拿下几何部分的分数，还能增强你在多步骤问题中的逻辑推理能力。
1. Basic Circle Terminology | 圆的基本术语
Before diving into the theorems, it is important to recall key terms. The centre is the fixed point from which all points on the circumference are equidistant. A radius (plural: radii) is a line segment from the centre to any point on the circumference. A chord is a line segment connecting two points on the circumference; a diameter is a special chord that passes through the centre. An arc is part of the circumference. A tangent is a straight line that touches the circle at exactly one point. A secant is a line that cuts the circle at two points. The angle formed by two radii is called the central angle, while an angle formed by two chords from a point on the circumference is called an inscribed angle.
在学习定理之前，先回顾基本术语。圆心是到圆周上所有点距离相等的定点。半径是从圆心到圆周上任意一点的线段。弦是连接圆周上两点的线段；直径是经过圆心的特殊弦。弧是圆周的一部分。切线是与圆仅在一个点相接触的直线。割线是与圆相交于两点的直线。由两条半径形成的角称为圆心角，而由圆周上一点出发的两条弦形成的角称为圆周角。
2. Angle at the Centre Theorem | 圆心角与圆周角定理
The angle subtended by an arc at the centre of the circle is twice the angle subtended by the same arc at any point on the remaining part of the circumference. If the central angle is θ, then the inscribed angle is θ/2. This theorem works provided both angles stand on the same arc.
同一段弧所对的圆心角等于该弧所对任一圆周角的两倍。若圆心角为 θ，则圆周角为 θ/2。使用该定理的前提是两个角立在相同的弧上。
Angle at centre = 2 × Angle at circumference
This is often the starting point for many IGCSE problems. You can recognise it when you see an angle at the centre and one at the edge sharing the same arc.
这是许多 IGCSE 题的起点。当你看到一个圆心角和一个在同弧上的圆周角时，就可以直接应用。
3. Angle in a Semicircle | 半圆上的圆周角
A triangle drawn in a semicircle using the diameter as its base will always have a right angle at the circumference. More formally: the angle in a semicircle is a right angle (90°). This is a special case of the angle at the centre theorem because the diameter creates a central angle of 180°, so the inscribed angle is 90°.
以直径为底边在半圆内画出的三角形，其圆周角总是直角（90°）。更规范的说法：半圆上的圆周角是直角。这是圆心角定理的特例，因为直径对应的圆心角为180°，故圆周角为90°。
∠ in a semicircle = 90°
WJEC exam questions often hide this theorem inside complex diagrams. Look for a triangle with one side as a diameter – the opposite angle must be 90°.
WJEC 考题常在复杂图形中隐藏该定理。寻找一条边为直径的三角形——其对顶点处的角必为90°。
4. Angles in the Same Segment | 同弧上的圆周角相等
All inscribed angles that stand on the same arc and are on the same side of the chord are equal. In other words, angles in the same segment of a circle are equal. If several points on the circumference lie on the same arc, the angles they subtend (with their vertices on that arc) are congruent.
所有立于同一弧上并在弦同侧的圆周角都相等。换言之，同一弓形内的角相等。如果圆周上有多个点位于同一段弧上，那么它们所对的圆周角（顶点在弧上）都是相等的。
∠APB = ∠AQB (same segment)
This theorem is extremely useful for proving that two angles are equal without measuring them. Always check if the vertices sit on the same arc.
该定理在无需度量的情况下证明两角相等非常有用。始终检查顶点是否在相同弧上。
5. Opposite Angles in a Cyclic Quadrilateral | 圆内接四边形的对角互补
A cyclic quadrilateral is a four‑sided figure with all vertices lying on the circumference of a circle. For any cyclic quadrilateral, the sum of each pair of opposite angles is 180° (they are supplementary). In diagram form: ∠A + ∠C = 180° and ∠B + ∠D = 180°.
圆内接四边形是指四个顶点均在同一个圆上的四边形。对于任何圆内接四边形，每组对角的和为180°（互补）。即 ∠A + ∠C = 180°，∠B + ∠D = 180°。
Opposite ∠s in cyclic quad add up to 180°
In WJEC exams, you may be given three angles and asked to find the fourth using this property. Alternatively, you might need to prove a quadrilateral is cyclic by showing opposite angles sum to 180°.
在 WJEC 考试中，可能会给出三个角，要求用该性质求第四个角。或者需要证明四边形为圆内接四边形，只需展示对角和为180°即可。
6. Tangent and Radius Theorem | 切线与半径垂直
A tangent to a circle is perpendicular to the radius drawn to the point of tangency. If a line is tangent to the circle at point P, and O is the centre, then OP is perpendicular to the tangent. This creates a right angle (90°) between the radius and the tangent.
圆的切线垂直于过切点的半径。若一直线在点 P 与圆相切，O 为圆心，则 OP 与该切线垂直。半径与切线之间形成一个直角（90°）。
Radius ⟂ Tangent at point of contact
This simple yet powerful theorem appears in almost every paper. Whenever you see a tangent, draw the radius to the contact point and mark the right angle. It often unlocks the first step of a multi‑step angle chase.
这个简单却强大的定理几乎出现在每份试卷中。无论何时看到切线，连接圆心和切点，标出直角。这往往是多步角度推算的第一步。
7. Tangents from an External Point | 圆外一点引出的两条切线
From an external point, two tangents can be drawn to a circle, and these tangents are equal in length. If point T is outside the circle and TP and TQ are tangents touching the circle at P and Q, then TP = TQ. Additionally, the line joining the external point to the centre bisects the angle between the tangents and also bisects the angle at the centre (∠PTO = ∠QTO).
从圆外一点可以引圆的两条切线，且这两条切线的长度相等。若点 T 在圆外，TP 和 TQ 为切线，切点分别为 P、Q，则 TP = TQ。另外，连接圆外点与圆心的直线平分两切线间的夹角，也平分圆心角（∠PTO = ∠QTO）。
Tangents from same point are equal
Exam questions often exploit the isosceles triangle formed by the two equal tangents. Expect to combine this with Pythagoras or trigonometry in WJEC problem‑solving questions.
试题常利用由两条等长切线构成的等腰三角形。在 WJEC 问题解决题中，可能需要将它与勾股定理或三角学结合。
8. Alternate Segment Theorem | 弦切角定理（交错弓形定理）
The angle between a tangent and a chord through the point of tangency is equal to the angle in the alternate segment. That is, ∠ between tangent and chord = angle in the opposite arc segment. If the chord divides the circle into two segments, the angle made by the tangent with the chord equals any angle subtended by the chord in the other segment.
切线与过切点的弦所夹的角，等于该弦所对的在交错弓形内的圆周角。即切线与弦的夹角 = 另一侧弓形内的圆周角。弦把圆分成两个弓形，切线与该弦的夹角等于弦所对的另一个弓形内的任意圆周角。
∠ between tangent and chord = ∠ in alternate segment
This is one of the most frequently misunderstood theorems. In WJEC IGCSE, it often appears when a tangent meets a chord, and you must identify the angle in the opposite segment. Practice recognising the ‘alternate’ segment to avoid confusion.
该定理是最容易被误解的定理之一。在 WJEC IGCSE 中，当切线与弦相交时经常出现，你必须识别出交错弓形内的角。练习辨认“交错”弓形，以避免混淆。
9. Perpendicular from Centre to a Chord | 弦心距定理
The perpendicular drawn from the centre of a circle to a chord bisects the chord. Conversely, the line joining the centre to the midpoint of a chord is perpendicular to the chord. This theorem creates right‑angled triangles inside the circle, which you can then solve using Pythagoras’ theorem.
从圆心向弦所作的垂线平分这条弦。反之，连接圆心和弦的中点的直线垂直于弦。该定理在圆内构造直角三角形，然后可以运用勾股定理来求解。
Centre to chord perpendicular → bisects chord
WJEC often embeds this theorem in questions involving the distance from the centre to a chord or finding the chord length given the radius and sagitta. Always draw the radius to one end of the chord to form a right triangle.
WJEC 常在涉及圆心到弦的距离或已知半径与矢高求弦长的问题中嵌入该定理。始终画一条半径到弦的一端，形成一个直角三角形。
10. Intersecting Chords Theorem (Extension) | 相交弦定理（拓展）
Although not always required for core IGCSE, some WJEC higher tier questions touch upon intersecting chords. For two chords intersecting inside a circle, the products of their segments are equal: AP × PB = CP × PD, where P is the intersection point. For two secants or chord extensions meeting outside, the same principle extends: PA × PB = PC × PD.
虽然核心 IGCSE 不总是要求，但部分 WJEC 更高层次的题目可能涉及相交弦定理。对于圆内相交的两条弦，它们被交点分成的两段乘积相等：AP × PB = CP × PD。对于圆外相交的割线或延长线，同样有 PA × PB = PC × PD。
AP × PB = CP × PD (intersecting chords)
This theorem is valuable for proving similarity and solving for unknown segment lengths. If you see two chords intersecting, consider using this relationship alongside other circle theorems.
该定理在证明相似性和求解未知线段长时很有价值。如果你看到两弦相交，考虑将此关系与其他圆定理结合使用。
11. Applying Circle Theorems to Multi‑Step Problems | 圆定理在多步问题中的应用
In WJEC IGCSE, a single question might require three or more circle theorems to find the solution. Begin by labelling all known angles on the diagram. Identify keywords: ‘diameter’ suggests the angle in a semicircle, ‘tangent’ points to the tangent‑radius perpendicular or alternate segment theorem, ‘cyclic quadrilateral’ means opposite angles sum to 180°. Build a chain of reasoning and clearly state each theorem as you use it.
在 WJEC IGCSE 中，一道题可能需要用到三个或更多的圆定理才能找到答案。首先在图上标注所有已知角度。抓关键词：“直径”提示半圆上的直角，“切线”指向切线与半径垂直或弦切角定理，“圆内接四边形”意味着对角互补。建立推理链，每应用一个定理都清楚地陈述出来。
Always write a short justification, e.g. ‘Angle ABC = 90° (angle in a semicircle)’. This not only gains you method marks but also prevents you from losing track of the logic.
始终写出简短的理由，例如 “∠ABC = 90°（半圆上的圆周角）”。这不仅帮你获得步骤分，还能防止逻辑混乱。
12. Revision Tips and Common Pitfalls | 复习提示与常见陷阱
When revising circle theorems, draw clear diagrams and label them with the theorem names. Memorise the exact wording, as WJEC mark schemes often require precise language. Common mistakes include confusing alternate segment with angles in the same segment, misidentifying the arc on which an angle stands, and forgetting to state that a line is a tangent before using tangent‑related theorems.
复习圆定理时，画出清晰示意图并以定理名称标注。熟记精确的表述，因为 WJEC 的评分标准通常要求准确的语言。常见错误包括：混淆弦切角定理和同弧圆周角定理，误判角所对的弧，以及在使用切线相关定理之前忘记声明某线是切线。
Do practise papers under timed conditions. Be systematic: always first look for a diameter (gives a right angle), then tangents (perpendicular radius or alternate segment), then cyclic quadrilaterals and so on. With consistent practice, circle theorem problems become a reliable source of marks.
在限时条件下做练习卷。要系统化：首先寻找直径（得直角），其次切线（垂直半径或弦切角），然后圆内接四边形等。通过持续练习，圆定理问题将成为可靠的得分点。
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