Circle Theorems | 圆中的角与线段定理

📚 Circle Theorems | 圆中的角与线段定理

Circle theorems describe relationships between angles, chords, tangents, and radii in a circle. They appear frequently in IGCSE Mathematics, especially in Paper 2 and Paper 4 geometry questions.

圆中的角与线段定理描述了圆内角、弦、切线与半径之间的关系。它们在 IGCSE 数学考试中频繁出现，尤其常见于 Paper 2 与 Paper 4 的几何题。

1. Key Language and Notation | 关键术语与记号

Before using circle theorems, you must recognise the main parts of a circle. A chord is a line segment joining two points on the circumference. A tangent is a straight line that touches the circle at exactly one point. An arc is part of the circumference between two points.

在使用圆定理之前，你必须先认识圆的主要部分。弦是连接圆上两点的线段；切线是与圆恰好相交于一点的直线；弧是圆上两点之间的圆周部分。

The angle subtended by an arc at the centre means the angle formed by the two radii to the endpoints of the arc. The same arc can also subtend an angle at the circumference. Use three-letter notation such as ∠AOB and ∠ACB to be precise.

弧所对的圆心角是指由连接弧两端点的两条半径形成的角。同一条弧也可以在圆周上对着一个圆周角。请使用三角标记法，例如 ∠AOB 和 ∠ACB，以保持清晰。

Common notation includes: O for the centre, r for the radius, AB for a chord, and P for an external point. Always label the centre first when reading a diagram.

常用记号包括：O 表示圆心，r 表示半径，AB 表示弦，P 表示圆外点。读图时请先标出圆心。

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2. Angle at the Centre | 圆心角定理

The angle at the centre of a circle is twice any angle at the circumference subtended by the same arc.

圆中，同一条弧所对的圆心角等于该弧所对任意圆周角的两倍。

If ∠AOB is the angle at the centre and ∠ACB is an angle at the circumference standing on the same arc AB, then:

如果 ∠AOB 是圆心角，∠ACB 是同一弧 AB 上的圆周角，那么：

∠AOB = 2 × ∠ACB

This theorem works when the angle at the centre is less than 180° or a reflex angle. Identify the arc first, then match the two angles to it.

该定理在圆心角小于 180° 或为优角时都成立。先确定弧，再找到对应的两个角。

For example, if ∠ACB = 32°, then ∠AOB = 64°.

例如，如果 ∠ACB = 32°，则 ∠AOB = 64°。

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3. Angle in a Semicircle | 半圆上的圆周角

An angle subtended by a diameter at the circumference is always a right angle.

由直径在圆周上所对的圆周角始终是直角。

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

如果 AB 是直径，C 是圆周上的任意一点，则 ∠ACB = 90°。

∠ACB = 90°

This is a special case of the angle at the centre theorem: the diameter gives a central angle of 180°, so the circumference angle is half of 180°.

这是圆心角定理的特殊情形：直径对应的圆心角为 180°，因此圆周角是其一半，即 90°。

Look for a diameter marked in the diagram. If you see 'angle in a semicircle', you can immediately mark a right angle and use Pythagoras or trigonometry.

注意图中是否标有直径。如果看到“半圆上的圆周角”，可立即标记直角，并使用勾股定理或三角函数。

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4. Angles in the Same Segment | 同弦上的圆周角

Angles subtended by the same chord or arc in the same segment are equal.

同一条弦或同一条弧在相同弓形中所对的圆周角相等。

If points C and D lie on the same side of chord AB, then ∠ACB = ∠ADB.

如果点 C 和点 D 位于弦 AB 的同一侧，则 ∠ACB = ∠ADB。

∠ACB = ∠ADB

Do not confuse this with the angle at the centre. Here both angles are at the circumference and share the same chord.

不要将此与圆心角定理混淆。此处两个角都在圆周上，并且共用同一条弦。

This theorem is useful for proving two angles equal in a diagram with several triangles built on the same base.

该定理常用于证明图中基于同一边的多个三角形中存在相等的角。

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5. Opposite Angles in a Cyclic Quadrilateral | 圆内接四边形对角定理

A cyclic quadrilateral is a four-sided shape with all four vertices on the circumference of a circle.

圆内接四边形是指四个顶点都在同一个圆上的四边形。

Opposite angles in a cyclic quadrilateral add up to 180°.

圆内接四边形的对角之和为 180°。

∠A + ∠C = 180° and ∠B + ∠D = 180°

This is often used to find a missing angle when three angles of the quadrilateral are known, or to prove that four points lie on a circle.

该定理常用于已知四边形三个角时求第四个角，或证明四点共圆。

exterior angle of a cyclic quadrilateral equals the interior opposite angle. For example, ∠CBE = ∠ADC when BCE is a straight line.

圆内接四边形的一个外角等于其内对角。例如，当 BCE 为直线时，∠CBE = ∠ADC。

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6. Tangent and Radius | 切线与半径

A tangent to a circle is perpendicular to the radius at the point of contact.

圆的切线在切点处垂直于过该切点的半径。

If line PT is tangent at T and O is the centre, then OT ⟂ PT, so ∠OTP = 90°.

如果直线 PT 在点 T 处与圆相切，O 为圆心，则 OT ⟂ PT，因此 ∠OTP = 90°。

OT ⟂ PT

This theorem creates a right-angled triangle whenever you join the centre to a tangent and to the point where the tangent touches the circle.

只要连接圆心与切点以及切线上的一点，该定理就能构造出一个直角三角形。

Use this right angle to apply Pythagoras' theorem, trigonometry, or angle sum in a triangle.

利用这个直角，可以应用勾股定理、三角函数或三角形内角和。

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7. Tangents from a Point | 圆外一点切线长定理

From an external point, the two tangent segments drawn to a circle are equal in length.

从圆外一点引圆的两条切线，它们的切线长相等。

If PA and PB are tangents from point P, then PA = PB.

如果 PA 和 PB 是从点 P 引出的两条切线，则 PA = PB。

PA = PB

ing the external point to the centre bisects the angle between the two tangents. Therefore ∠APO = ∠BPO.

此外，连接圆外点与圆心的直线平分两条切线之间的夹角，因此 ∠APO = ∠BPO。

This theorem is useful in problems with a point outside a circle and two tangent lines. Mark equal lengths and equal angles immediately.

该定理常用于圆外一点有两条切线的题目。请立即标出相等线段与相等角度。

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8. Alternate Segment Theorem | 弦切角定理

The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.

切线与经过切点的弦所夹的角，等于该角所对应的另一侧圆周角，即弦切角定理。

If line PT is a tangent at T and chord TU is drawn, then ∠PTU = ∠TVU where V is any point on the circumference in the alternate segment.

如果 PT 是在点 T 处的切线，TU 是经过 T 的弦，则 ∠PTU = ∠TVU，其中 V 是另一侧圆周上的任意点。

∠PTU = ∠TVU

Students often overlook this theorem because it does not create a right angle directly. Identify the tangent-chord angle and then find the angle in the opposite segment.

学生常因该定理不直接产生直角而忽略它。先找出切线与弦的夹角，再找到另一侧对应的圆周角。

The alternate segment theorem can be used to prove equal angles in diagrams with tangents and chords, especially when no radius is drawn.

弦切角定理可用于证明切线与弦图形中的等角，尤其是在未画出半径时。

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9. Chords and Perpendicular Bisectors | 弦与垂直平分线定理

The perpendicular from the centre

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