📚 Circle Theorems | 圆的定理
The circle is perhaps the most elegant shape in geometry, and IGCSE Mathematics devotes a complete module to its special angle properties. These properties, called 'circle theorems', let us calculate angles without any measurement, using only a few powerful rules. This article explains every theorem you need, the reasoning behind each one, and the exact language examiners want to see in your answer.
圆或许是几何中最优美的图形，而 IGCSE 数学用整整一个模块来讲解它的特殊角度性质。这些性质统称为“圆的定理”，它让我们不借助任何测量，仅用几条强有力的规则就能求出角度。本文会讲解你需要的每一条定理、每一条定理背后的推理，以及考官希望你在答卷中写出的准确表述。
1. Circle Vocabulary | 圆的常用术语
Before applying the theorems, you must be able to name the parts of a circle. Every theorem in this chapter is built from these five words: centre, radius, chord, tangent and arc.
在套用定理之前，你必须能准确说出圆各部分的名称。本章的所有定理都由五个核心词构成：圆心、半径、弦、切线和弧。
Circumference: the whole distance around the circle; C = πd = 2πr.
周长：绕圆一整圈的长度；C = πd = 2πr。
Chord: a straight line joining two points on the circle. The diameter is the longest chord and passes through the centre.
弦：连接圆上两点的直线。直径是最长的弦，并且经过圆心。
Arc: part of the circumference. A minor arc is shorter than a semicircle; a major arc is longer.
弧：圆周的一部分。劣弧短于半圆，优弧长于半圆。
Segment: the region between a chord and its arc.
弓形：一条弦与其对应弧之间的区域。
Tangent: a straight line that touches the circle at exactly one point.
切线：与圆恰好只有一个公共点的直线。
Angle at the centre: the angle formed at O by two radii, e.g. ∠AOB.
圆心角：两条半径在圆心 O 处构成的角，例如 ∠AOB。
Angle at the circumference: the angle formed on the circle by two chords, e.g. ∠APB.
圆周角：两条弦在圆周上构成的角，例如 ∠APB。
Circumference C = πd = 2πr, Area A = πr²
2. Theorem 1: Angle at the Centre | 定理一：圆心角是圆周角的两倍
If A and B are two points on a circle, and P is any point on the same arc, then the angle at the centre is exactly twice the angle at the circumference.
若 A、B 是圆上的两点，P 是同一条弧上的任意一点，那么圆心角正好是圆周角的两倍。
∠AOB = 2 × ∠APB
Why does this work? osceles triangle, the exterior angle at O equals twice the base angle, so the two parts of ∠AOB are 2x and 2y. Hence ∠AOB = 2x + 2y = 2(x + y) = 2∠APB.
为什么成立呢？注意 OA = OP = OB，因为它们都是半径。因此三角形 OAP 与 OPB 都是等腰三角形。设两个等腰三角形的底角分别为 x 和 y。在等腰三角形中，O 处的外角等于底角的两倍，所以 ∠AOB 的两部分分别是 2x 和 2y。因此 ∠AOB = 2x + 2y = 2(x + y) = 2∠APB。
In your IGCSE answer, always write the reason: 'angle at the centre is twice the angle at the circumference'. This exact sentence earns the reasoning mark.
在 IGCSE 答卷中，一定要写出理由：“圆心角是圆周角的两倍”。这个标准表述能为你赢得推理分。
3. Theorem 2: Angles in the Same Segment | 定理二：同弧上的圆周角相等
lie in the same segment and are equal. If P and Q are two points on the same arc AB, then ∠APB = ∠AQB.
同一条弦在同一侧所张出的两个圆周角位于同一个弓形内，它们相等。若 P、Q 是同一条弧 AB 上的两点，则 ∠APB = ∠AQB。
∠APB = ∠AQB
a direct consequence of Theorem 1: both angles are equal to half of the same central angle ∠AOB. The converse is also true: if ∠APB = ∠AQB, then A, B, P and Q lie on the same circle, so the four points are concyclic.
该定理是定理一的直接推论：这两个角都等于同一个圆心角 ∠AOB 的一半。其逆命题也成立：若 ∠APB = ∠AQB，则 A、B、P、Q 四点共圆。
4. Theorem 3: Angle in a Semicircle | 定理三：半圆所对的圆周角是直角
Because a diameter is a straight line passing through the centre, the angle at the centre subtended by a diameter is 180°. Half of 180° is 90°, so every angle subtended by a diameter at the circumference is a right angle.
因为直径是经过圆心的直线，直径所对应的圆心角为 180°。180° 的一半是 90°，因此直径在圆周上所张出的任意圆周角都是直角。
∠APB = 90° when AB is a diameter
. Exam questions often combine this with Pythagoras: a² + b² = c², to find a missing side.
这意味着：由直径两个端点与圆上第三点组成的三角形一定是直角三角形。考题常将此定理与勾股定理 a² + b² = c² 结合，用来求未知边长。
5. Theorem 4: Cyclic Quadrilaterals | 定理四：圆内接四边形对角互补
A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the same circle. In any cyclic quadrilateral, opposite angles add up to 180°.
圆内接四边形是指四个顶点都在同一个圆上的四边形。在任意圆内接四边形中，对角之和为 180°。
∠a + ∠c = 180°, ∠b + ∠d = 180°
There is also a very useful consequence: the exterior angle of a cyclic quadrilateral equals the opposite interior angle. For example, if one side is extended, the outside angle equals the interior angle at the opposite vertex.
这里还有一个非常有用的推论：圆内接四边形的外角等于它的内对角。例如，将一条边延长后，所得的外角等于对角顶点的内角。
If you need to prove that four points lie on a circle, you can show that a pair of opposite angles sums to 180°, or that one exterior angle equals the opposite interior angle.
如果你需要证明四点共圆，只需证明其中一对对角之和为 180°，或证明一个外角等于其内对角。
6. Theorem 5: Alternate Segment Theorem | 定理五：弦切角定理（交替弓形）
If a tangent touches the circle at T and a chord TQ is drawn from T, then the angle between the tangent and the chord TQ equals the angle subtended by that chord in the alternate segment.
若切线与圆相切于点 T，从 T 作弦 TQ，那么切线与弦 TQ 的夹角等于该弦所对“交替弓形”内的圆周角。
Angle between tangent and TQ = ∠TRQ
Here R is any point on the circle on the opposite side of chord TQ from the tangent. The word 'alternate' means 'the other side': imagine the tangent and the chord cutting the circle into two segments, and look at the segment on the other side from the angle you are measuring.
其中 R 是圆上位于弦 TQ 的另一侧（与切线相对的一侧）的任意一点。“交替”的意思是“另一侧”：想象切线与弦把圆分成两个弓形，你要找的是与你所测角相对的那个弓形内的角。
7. Theorem 6: Tangent Properties | 定理六：切线的性质
A tangent is always perpendicular to the radius drawn to the point of contact. Also, two tangents drawn from the same external point to a circle are equal in length.
切线在切点处总是垂直于过切点的半径。此外，从同一个圆外一点引出的两条切线长度相等。
