IGCSE Mathematics: Circle Theorems | 圆的基本定理

📚 IGCSE Mathematics: Circle Theorems | 圆的基本定理

Circle theorems are a core part of IGCSE Mathematics. They describe the relationships between angles, chords, tangents, and arcs in a circle. Mastering these theorems allows students to solve complex geometry problems with elegant reasoning.

圆的基本定理是 IGCSE 数学的核心内容。它们描述了圆中角、弦、切线和弧之间的关系。掌握这些定理，学生就能用简洁的推理解决复杂的几何问题。

1. Key Parts of a Circle | 圆的基本元素

A circle has several important parts. The centre is the middle point, the radius is a line from the centre to the circumference, and the diameter is a chord that passes through the centre, equal to two radii. An arc is part of the circumference, and a chord is a line segment with both endpoints on the circle. A sector is the region between two radii and an arc, while a segment is the region between a chord and an arc.

圆有几个重要的元素。圆心是中间的点，半径是从圆心到圆周的线段，直径是经过圆心的弦，等于两个半径的长度。弧是圆周的一部分，弦是两端都在圆上的线段。扇形是两个半径与一段弧围成的区域，而弓形是一条弦与一段弧围成的区域。

These basic definitions are essential before applying theorems. For example, the angle subtended by a diameter is always a right angle, which relies on the definition of a diameter.

这些基本定义是运用定理之前必须掌握的。例如，直径所对的圆周角总是直角，这依赖于直径的定义。

2. Angle at the Centre | 圆心角定理

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 is the most important circle theorem.

同弧所对的圆心角等于该弧在圆周上任一点所对的圆周角的两倍。这是最重要的圆定理。

∠AOC = 2 × ∠ABC

Here, points A, B and C lie on the circle, O is the centre, and B is a point on the circumference. The arc AC subtends the angle at the centre and the angle at the circumference.

这里 A、B、C 在圆上，O 是圆心，B 是圆周上的一点。弧 AC 分别对圆心角和圆周角。

3. Angle in a Semicircle | 直径所对的圆周角

If a triangle is inscribed in a circle with one side as the diameter, then the angle opposite the diameter is a right angle. In other words, the angle subtended by a diameter is 90°.

如果一个三角形内接于圆，且一边是直径，那么直径所对的角是直角。换句话说，直径所对的圆周角等于 90°。

∠ACB = 90°

This theorem is often used to prove that a triangle is right-angled or to construct right angles using a circle.

这个定理常用于证明三角形是直角三角形，或利用圆构造直角。

4. Angles in the Same Segment | 同弧上的圆周角

Angles in the same segment are equal. That means if two points D and E are on the same side of a chord AB, then the angles ∠ADB and ∠AEB are equal.

同弧上的圆周角相等。也就是说，如果 D 和 E 在弦 AB 的同一侧，那么 ∠ADB 和 ∠AEB 相等。

∠ADB = ∠AEB

This theorem is particularly useful when solving questions that involve multiple points on the same arc.

这个定理在解决涉及同一条弧上多个点的问题时特别有用。

5. Angles in a Cyclic Quadrilateral | 圆内接四边形

A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the circumference of a circle. The opposite angles of a cyclic quadrilateral are supplementary, meaning they sum to 180°.

圆内接四边形是四个顶点都在同一个圆上的四边形。圆内接四边形的对角互补，即它们的和为 180°。

∠A + ∠C = 180°, ∠B + ∠D = 180°

This property also leads to the exterior angle of a cyclic quadrilateral being equal to the interior opposite angle.

这一性质还推出圆内接四边形的外角等于内对角。

6. Perpendicular from Centre to Chord | 圆心到弦的垂线

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.

从圆心到弦作垂线，垂线平分这条弦。反之，连接圆心与弦中点的直线垂直于该弦。

If OM ⊥ AB, then AM = MB

This theorem is fundamental for calculating lengths in circles using Pythagoras’ theorem.

这个定理是利用勾股定理计算圆中长度的基础。

7. Chords of Equal Length | 等弦与等弧

Equal chords are equidistant from the centre. Conversely, chords equidistant from the centre are equal. Equal chords also subtend equal arcs and equal angles at the centre.

相等的弦到圆心的距离相等。反之，到圆心距离相等的弦也相等。等弦所对的弧相等，所对的圆心角也相等。

If chord AB = chord CD, then ∠AOB = ∠COD

This relationship helps to prove symmetry properties within a circle.

这个关系有助于证明圆内的对称性质。

8. Tangent to a Circle | 圆的切线

A tangent to a circle is a straight line that touches the circle at exactly one point. The angle between a tangent and the radius drawn to the point of contact is 90°.

圆的切线是与圆只有一个公共点的直线。切线与过切点的半径所成的角是 90°。

OT ⟂ PT

Two tangents drawn from an external point to a circle are equal in length. Thus, PA = PB, where P is the external point and A, B are the points of contact.

从圆外一点引圆的两条切线，它们的长度相等。因此，若 P 是圆外一点，A、B 是切点，则 PA = PB。

9. Alternate Segment Theorem | 弦切角定理

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment of the circle. This theorem is also called the tangent-chord theorem.

切线与过切点的弦所成的角，等于该弦所对的、位于切线另一侧的圆周角。这个定理也叫弦切角定理。

∠PTA = ∠TBA

Here, PT is a tangent at point T, and chord TA subtends the angle at B on the opposite arc.

这里 PT 是圆在点 T 处的切线，弦 TA 所对的角在另一弧上的 B 点处。

10. Intersecting Chords and Secants | 相交弦与割线

When two chords intersect inside a circle, the product of the two segments of one chord equals the product of the two segments of the other chord. This is known as the intersecting chords theorem.

当两条弦在圆内相交时，其中一条弦被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。这称为相交弦定理。

PA × PB = PC × PD

The same idea extends to secants meeting outside the circle: the product of the external segment and the whole secant is equal for both secants. For a tangent and a secant, the tangent length squared equals the product of the external segment and the whole secant.

同样的想法也适用于两条割线在圆外交于一点的情况：外部线段与整条割线的乘积相等。对于切线与割线，切线长的平方等于外部线段与整条割线的乘积。

11. Worked Example | 例题解析

In the diagram below, A, B, C and D lie on a circle, O is the centre, ∠AOB = 80°, and ∠COD = 100°. Find ∠ACB.

如下图所示，A、B、C、D 在圆上，O 是圆心，∠AOB = 80°，∠COD = 100°。求 ∠ACB。

Solution: Since ∠AOB is the angle at the centre subtending arc AB, the angle at the circumference subtending the same arc is half as large. Therefore ∠ACB = ½ × ∠AOB = 40°. Note that point D is irrelevant for this angle.

解答：因为 ∠AOB 是弧 AB 所对的圆心角，所以同弧所对的圆周角是它的一半。因此 ∠ACB = ½ × ∠AOB = 40°。注意点 D 对这个角无关。

12. Exam Tips | 考试提示

In IGCSE exams, always cite the theorem you are using, for example ‘angle at the centre is twice the angle at the circumference’. Show each step clearly and use correct notation. When solving problems, mark any known angles on the diagram first.

在 IGCSE 考试中，一定要写出你所用的定理，例如“圆心角是圆周角的两倍”。每一步都要写清楚并使用正确的符号。解题时，先在图上标出已知角。

Practice combining multiple theorems. Many exam questions require two or more circle theorems to find a single angle.

练习将多个定理结合使用。许多考试题目需要两个或更多圆定理才能求出一个角。

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IGCSE Mathematics: Circle Theorems | 圆的基本定理