📚 Circle Theorems for IGCSE Mathematics | IGCSE 数学圆定理精讲
Circle theorems describe angle and length relationships in circles. They appear frequently in IGCSE Mathematics, especially in geometry questions that require you to find missing angles. A strong understanding of these theorems will help you solve problems faster and justify your answers clearly. In this article, we review the main circle theorems, give simple examples, and share exam tips.
圆定理描述圆内角度和线段长度之间的关系。它们在 IGCSE 数学中经常出现，尤其是在求未知角的几何题中。扎实掌握这些定理能帮助你更快解题并清晰地写出推理过程。本文将复习主要圆定理，给出简单示例，并分享应试技巧。
1. Angle at the Centre Theorem | 圆心角定理
he same arc. In notation, if A, B and C lie on the circumference and O is the centre, then ∠AOB = 2 × ∠ACB.
当圆心角和圆周角由同一条弧所对时，圆心角等于圆周角的两倍。用符号表示，如果 A、B、C 在圆周上，O 为圆心，那么 ∠AOB = 2 × ∠ACB。
questions where the centre is given but the diagram is not drawn to scale. For example, if ∠ACB = 34°, then ∠ °. Always identify the arc first and check that both angles are subtended by the same arc.
该定理常用于已知圆心但图形未按比例绘制的考题中。例如，如果 ∠ACB = 34°，那么 ∠AOB = 68°。解题时先识别同一条弧，并确认两个角都由这条弧所对。
2. Angle in a Semicircle | 半圆上的圆周角
The angle subtended by a diameter at the circumference is always 90°. This is a special case of the angle at the centre theorem because the diameter forms an angle of 180° at the centre.
直径在圆周上所对的圆周角始终等于 90°。这是圆心角定理的特殊情况，因为直径在圆心处形成的角为 180°。
This theorem is extremely useful when you see a triangle inside a circle with one side as the diameter. You can immediately mark the opposite angle as a right angle, which often allows simple subtraction to find other angles.
当你看到圆内接三角形的一条边是直径时，这个定理非常有用。你可以立即把对边角标为直角，通常再用简单减法就能求出其他角。
3. Angles in the Same Segment | 同弓形内的圆周角
Angles subtended by the same chord in the same segment of a circle are equal. This means if two points are fixed on the circumference and the angle is drawn from a third point on the same arc side, all such angles are equal.
同一条弦在同一个弓形内所对的圆周角相等。也就是说，如果圆周上有两个固定点，从同一弧侧的第三个点作圆周角，所有这些圆周角都相等。
In complex diagrams, look for a chord that is shared by two angles facing the same side of the chord. You can then set the angles equal and solve for unknown variables.
在复杂图形中，寻找一条被两个位于弦同侧的角所共用的弦。然后你可以令这两个角相等，从而求出未知量。
4. Cyclic Quadrilaterals | 圆内接四边形
A cyclic quadrilateral is a four-sided shape with all four vertices lying on the circumference of a circle. The key theorem is that opposite angles add up to 180°.
圆内接四边形是指四个顶点都在圆周上的四边形。其关键定理是对角之和等于 180°。
ertices and write equations such as angle A + angle C = 180° and angle B + angle D = 180°.
这个性质也可以反过来用：如果一个四边形的对角之和等于 180°，那么它内接于圆。在考试题中，先标出顶点，再写出方程，例如角 A + 角 C = 180° 和角 B + 角 D = 180°。
5. Tangent-Radius Theorem | 切线与半径定理
A tangent is a straight line that touches the circle at exactly one point. The tangent is perpendicular to the radius drawn to the point of contact. Therefore, the angle between a tangent and the radius is 90°.
切线是与圆只有一个公共点的直线。切线与经过切点的半径垂直。因此，切线与半径之间的夹角为 90°。
This creates right-angled triangles when a tangent and a radius are shown together. Use Pythagoras' theorem or trigonometric ratios if a length is required.
当切线和半径同时出现时，会形成直角三角形。如果需要求长度，可使用勾股定理或三角比。
6. Tangents from an External Point | 圆外一点引切线
P is an external point and PA and PB are tangents to the circle at A and B, then PA = PB.
如果从圆外同一点向圆作两条切线，那么这两条切线的长度相等。例如，P 是圆外一点，PA 和 PB 是分别切圆于 A 和 B 的切线，则 PA = PB。
This theorem is often combined with the tangent-radius theorem. The line joining the external point to the centre bisects the angle between the two tangents. These facts help solve length and angle problems involving tangents.
这个定理常与切线和半径定理结合使用。连接圆外点与圆心的直线平分两条切线的夹角。这些结论有助于解决涉及切线的长度和角度问题。
7. Alternate Segment Theorem | 交替线段定理
The angle between a tangent and a chord at the point of contact is equal to the angle subtended by the chord in the alternate segment of the circle.
切线与过切点的弦之间所夹的角，等于弦在圆的交替线段中所对的圆周角。
This theorem can be confusing because the equal angles are on opposite sides of the chord. A useful strategy is to highlight the chord and the tangent, then locate the angle in the opposite segment that sits between the chord and the circumference.
这个定理容易混淆，因为相等的角位于弦的两侧。一个有用的方法是先标出弦和切线，然后在另一侧的弓形内找到位于弦和圆周之间的角。
8. Intersecting Chords Theorem | 相交弦定理
rd. If P is the intersection point and chords AB and CD meet at P, then PA × PB = PC × PD.
如果两条弦在圆内相交，一条弦被交点分成的两段长度之积等于另一条弦被分成的两段长度之积。如果 P 是交点，弦 AB 和 CD 相交于 P，那么 PA × PB = PC × PD。
× PD, where P is the external point and the points are measured along the secants from P.
当两条割线在圆外相交时，同样的结论也成立：PA × PB = PC × PD，其中 P 是圆外交点，长度从 P 沿割线度量。
theorem is useful for both finding unknown lengths and proving proportional relationships.
一个简单例子：如果 PA = 3，PB = 4，PC = 6，则 PD = (3 × 4) ÷ 6 = 2。这个定理既可用于求未知长度，也可用于证明比例关系。
9. Applying Theorems in Multi-Step Problems | 多步问题中的应用
IGCSE questions often ask you to find an unknown angle using two or more theorems. Start by marking all angles that can be found directly: right angles from
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