Circle Theorems | 圆定理

📚 Circle Theorems | 圆定理

Circle theorems are a set of fundamental rules that describe the relationships between angles, chords, tangents, and arcs in a circle. Mastering these theorems is essential for success in IGCSE Mathematics, as they appear frequently in both Paper 2 and Paper 4 examinations across all major exam boards including CAIE, Edexcel, and AQA.

圆定理是一组描述圆中角、弦、切线和弧之间基本关系的法则。掌握这些定理对于在 IGCSE 数学中取得好成绩至关重要，因为它们在卷二和卷四考试中频繁出现，适用于 CAIE、爱德思（Edexcel）和 AQA 等所有主要考试局。

1. The Angle at the Centre Theorem | 圆心角定理

The angle subtended by an arc at the centre of a circle is exactly twice the angle subtended by the same arc at any point on the circumference. This is one of the most powerful and frequently tested circle theorems in the IGCSE syllabus.

圆的一条弧在圆心处所张的角，恰好是同一弧在圆周上任意一点处所张的角的两倍。这是 IGCSE 考纲中最重要且最常被考查的圆定理之一。

The angle at the centre = 2 × the angle at the circumference

圆心角 = 2 × 圆周角

For example, if a chord AB subtends an angle of 30° at a point C on the circumference, then the angle subtended by AB at the centre O is 60°. This theorem works for both major and minor arcs. However, you must ensure that the angle at the centre and the angle at the circumference are standing on the same arc.

例如，如果弦 AB 在圆周上一点 C 处张成 30° 的角，那么 AB 在圆心 O 处张成的角就是 60°。这一定理对优弧和劣弧均适用。但是，你必须确保圆心角和圆周角对应的是同一条弧。

2. Angles in the Same Segment | 同弓形内角相等

Angles subtended by the same chord at the circumference are equal, provided that they lie in the same segment of the circle. In other words, if two points are located on the same side of a chord and both lie on the circumference, the angles they form with the endpoints of that chord are equal.

同一条弦在圆周上张成的角相等，前提是这些角位于圆的同一弓形内。换句话说，如果两个点位于同一条弦的同侧且都在圆周上，它们与该弦端点形成的角相等。

Consider a circle with chord PQ. If points R and S lie on the same arc of chord PQ, then ∠PRQ = ∠PSQ. This theorem is particularly useful for proving other geometric relationships and solving angle-chasing problems where multiple points lie on a common circle.

考虑一个具有弦 PQ 的圆。如果点 R 和 S 位于弦 PQ 的同一侧弧上，则 ∠PRQ = ∠PSQ。这一定理在证明其他几何关系以及解决多个点位于同一圆上的角度推理问题中特别有用。

3. The Angle in a Semicircle | 半圆内的角

An angle subtended by a diameter at the circumference is always a right angle, measuring exactly 90°. This is a special case of the angle at the centre theorem: the angle at the centre for a diameter is a straight angle of 180°, and half of 180° is 90°.

直径在圆周上所张的角始终是直角，恰好为 90°。这是圆心角定理的一个特例：直径在圆心处所张的角是 180° 的平角，而 180° 的一半就是 90°。

Theorem statement: If AB is a diameter of a circle and C is any point on the circumference, then ∠ACB = 90°.

定理表述： 如果 AB 是圆的直径，C 是圆周上的任意一点，则 ∠ACB = 90°。

This theorem is often applied in reverse as well: if a triangle is inscribed in a circle with one side being the diameter, then the triangle must be right-angled. This connection is frequently tested in IGCSE questions that combine circle geometry with the Pythagorean theorem.

这个定理经常被反向运用：如果一个三角形内接于圆且有一条边是直径，那么这个三角形必为直角三角形。这一联系常出现在将圆几何与勾股定理相结合的 IGCSE 题目中。

4. Cyclic Quadrilateral Theorem | 圆内接四边形定理

A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the circumference of a circle. The opposite angles of a cyclic quadrilateral sum to 180°. This happens because each pair of opposite angles stands on complementary arcs of the circle.

圆内接四边形是指四个顶点都在同一个圆的圆周上的四边形。圆内接四边形的对角之和为 180°。这是因为每一对对对角对应的是圆中互补的弧。

∠A + ∠C = 180° and ∠B + ∠D = 180°

∠A + ∠C = 180° 且 ∠B + ∠D = 180°

In addition, an exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. For example, if side BC is extended to point E, then ∠DCE = ∠BAD. This property is extremely valuable in solving multi-step angle problems, especially those involving parallel lines and triangles.

此外，圆内接四边形的一个外角等于其内对角。例如，如果将边 BC 延长到点 E，则 ∠DCE = ∠BAD。这个性质在解决多步骤角度问题中极为有用，尤其是涉及平行线和三角形的问题。

5. Tangent and Radius Theorem | 切线与半径定理

The tangent to a circle at any point is perpendicular to the radius drawn to the point of contact. If OT is a radius and the tangent touches the circle at point T, then the angle between OT and the tangent line is 90° at point T.

圆在任意一点的切线与经过切点的半径互相垂直。如果 OT 是半径，切线与圆相切于点 T，则 OT 与切线在点 T 处的夹角为 90°。

This theorem leads to several important consequences that are frequently examined:

这一定理引出了几个常被考查的重要推论：

The perpendicular drawn from the centre to a tangent line meets the circle exactly at the point of tangency.

从圆心向切线所作的垂线与圆相交的位置恰好就是切点。
If two tangents are drawn to a circle from an external point, the line segment joining the centre to that external point bisects the angle between the two tangents.

如果从圆外一点向圆作两条切线，连接圆心与该圆外点的线段平分这两条切线之间的夹角。

6. Alternate Segment Theorem | 弦

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Circle Theorems | 圆定理