GCSE Edexcel Maths: Circle Geometry Key Points | GCSE Edexcel 数学：圆周运动考点精讲

📚 GCSE Edexcel Maths: Circle Geometry Key Points | GCSE Edexcel 数学：圆周运动考点精讲

Circle geometry is a core topic in the GCSE Edexcel Mathematics syllabus. It involves understanding the properties of circles, including chords, tangents, arcs, sectors, and the angles formed by lines intersecting on, inside, or outside a circle. Mastering these circle theorems not only helps you solve geometric problems efficiently but also builds a foundation for further study in trigonometry and coordinate geometry. This article covers the essential circle theorems, their proofs where relevant, and typical exam-style applications.

圆周运动在 GCSE Edexcel 数学大纲中是核心考点。它涉及对圆的性质的理解，包括弦、切线、弧、扇形，以及由相交于圆上、圆内或圆外的直线所形成的角。掌握这些圆定理不仅能帮助你高效解决几何问题，也为进一步学习三角学和坐标几何打下基础。本文涵盖关键的圆定理、必要的证明以及典型的考试题型应用。

1. Basic Circle Terminology | 圆的基本术语

A circle is the set of all points in a plane that are at a fixed distance (radius) from a fixed point (centre). Key terms include radius (r), diameter (d = 2r), chord (a line segment whose endpoints lie on the circle), tangent (a line that touches the circle at exactly one point), arc (a part of the circumference), sector (the region enclosed by two radii and an arc), and segment (the region enclosed by a chord and an arc).

圆是平面上到一个定点（圆心）距离等于定长（半径）的所有点的集合。关键术语包括半径（r）、直径（d = 2r）、弦（端点都在圆上的线段）、切线（与圆恰好只有一个公共点的直线）、弧（圆周长的一部分）、扇形（由两条半径和一段弧围成的区域）以及弓形（由一条弦和一段弧围成的区域）。

The angle at the centre is the angle formed by two radii. The angle at the circumference is formed by two chords that meet on the circle. These angles are fundamental to many circle theorems.

圆心角是由两条半径所夹的角。圆周角是由在圆上相交的两条弦所夹的角。这些角在许多圆定理中都是基础。

2. Perpendicular Bisector of a Chord | 弦的垂直平分线

The perpendicular from the centre of a circle to a chord bisects the chord. Conversely, the line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord. This means that if you draw a line from the centre to the midpoint of a chord, it forms a right angle (90°) with the chord, and splits the chord into two equal lengths.

从圆心引向弦的垂线平分这条弦。反过来，连接圆心和弦的中点的直线垂直于这条弦。这意味着如果你从圆心到弦的中点画一条线，它会与弦形成直角（90°），并将弦分成相等的两段。

This theorem is often used to find the distance from the centre to a chord, or to calculate the length of a chord when the radius and distance from the centre are known. For example, if a circle has radius 5 cm and a chord is 6 cm long, the distance from the centre to the chord can be found using Pythagoras’ theorem: perpendicular distance = √(5² – 3²) = 4 cm.

这个定理常用于求圆心到弦的距离，或在已知半径和弦心距时计算弦长。例如，若圆半径为 5 cm，弦长为 6 cm，圆心到弦的距离可用勾股定理求得：垂直距离 = √(5² – 3²) = 4 cm。

3. Tangent-Radius Theorem | 切线半径定理

A tangent to a circle is perpendicular to the radius drawn to the point of contact. If a line touches a circle at point P, and O is the centre, then OP is perpendicular to the tangent at P. This gives a right angle (90°).

圆的切线垂直于过切点的半径。如果一条直线在点 P 处与圆相切，O 为圆心，那么 OP 垂直于在 P 处的切线。这构成一个直角（90°）。

This theorem is particularly useful when solving problems involving tangents from an external point. The two tangent segments drawn from an external point to a circle are equal in length. So from point T outside the circle, the two tangents TA and TB (A and B are points of contact) satisfy TA = TB. This creates congruent right-angled triangles OAT and OBT.

这个定理在解决涉及外点切线的问题时特别有用。从圆外一点引圆的两条切线，它们的切线长相等。因此，从圆外一点 T 引出的两条切线 TA 和 TB（A、B 为切点）满足 TA = TB。这构成了全等的直角三角形 OAT 和 OBT。

4. Angle at the Centre and Circumference | 圆心角与圆周角

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 remaining part of the circumference. That is, if an arc AB subtends angle AOB at the centre and angle ACB at the circumference, then ∠AOB = 2 × ∠ACB.

同一段弧所对的圆心角等于它所对的圆周角的两倍。即，如果弧 AB 所对的圆心角为 ∠AOB，所对的圆周角为 ∠ACB，那么 ∠AOB = 2 × ∠ACB。

This is one of the most commonly tested circle theorems. It can be proved using the fact that the exterior angle of a triangle equals the sum of the two opposite interior angles, and the radii create isosceles triangles. In exams, look for the ‘bow-tie’ or ‘arrowhead’ shape where the centre is involved.

这是最常见的圆定理之一。可以利用三角形外角等于两个不相邻内角之和，以及半径构成等腰三角形来证明。在考试中，寻找包含圆心的“蝴蝶结”或“箭头”形状。

5. Angles in the Same Segment | 同一圆弧上的圆周角相等

Angles in the same segment of a circle are equal. This means that if two angles are subtended by the same chord (or arc) and their vertices lie on the same side of the chord, then they are equal. For example, chord AB defines a segment; any angle ACB and angle ADB (with C and D on the same arc) are equal.

同一段弧所对的圆周角相等。这意味着如果两个角都对着同一条弦（或弧），并且它们的顶点位于弦的同侧，那么这两个角相等。例如，弦 AB 定义了一个弓形；任何一个 ∠ACB 和 ∠ADB（C 和 D 在同一段弧上）相等。

This theorem follows directly from the angle at the centre theorem: since both angles are half of the same central angle, they must be equal. Exam questions often hide this by drawing multiple chords, so learners must identify the common chord and the relevant arc.

这个定理直接由圆心角定理推出：既然两个角都是同一个圆心角的一半，它们必然相等。考试题目常常通过画多条弦来隐藏这一关系，考生需要识别出公共弦和相关的弧。

6. Angle in a Semicircle | 半圆上的圆周角

The angle subtended by a diameter (or a semicircle) at any point on the circumference is a right angle (90°). If AB is a diameter of the circle and C is any point on the circumference (other than A and B), then ∠ACB = 90°.

直径（或半圆）所对的圆周角是直角（90°）。如果 AB 是圆的直径，C 是圆周上任意一点（不同于 A 和 B），那么 ∠ACB = 90°。

This is a special case of the angle at the centre theorem: the central angle is 180° for a diameter, so the inscribed angle is 90°. This theorem is often used to prove that a triangle inscribed in a semicircle is right-angled, or to spot right angles in composite shapes involving circles.

这是圆心角定理的一个特例：直径对应的圆心角为 180°，所以圆周角为 90°。这个定理常用于证明内接于半圆的三角形是直角三角形，或者在涉及圆的组合图形中识别直角。

7. Cyclic Quadrilateral | 圆内接四边形

A cyclic quadrilateral is a four-sided figure whose vertices all lie on a single circle. The opposite angles of a cyclic quadrilateral sum to 180° (they are supplementary). If quadrilateral ABCD is cyclic, then ∠A + ∠C = 180° and ∠B + ∠D = 180°.

圆内接四边形是指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，即对角之和为 180°。如果四边形 ABCD 是圆内接四边形，那么 ∠A + ∠C = 180° 且 ∠B + ∠D = 180°。

The exterior angle of a cyclic quadrilateral equals the interior opposite angle. For example, if side AD is extended to point E, then ∠CDE = ∠ABC. This property is useful for proof questions and angle chasing.

圆内接四边形的外角等于内对角。例如，如果边 AD 延长到点 E，那么 ∠CDE = ∠ABC。这个性质在证明题和角度推理中非常有用。

8. Alternate Segment Theorem | 切线与弦的夹角定理

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. If a tangent at point A touches the circle, and AB is a chord, then the angle between the tangent and chord AB equals the angle subtended by the chord AB in the opposite segment (i.e., angle ACB where C is any point on the circle on the other side of AB).

切线与经过切点的弦所夹的角等于该弦在另一侧弓形中所对的圆周角。如果在点 A 处有一条切线，AB 是一条弦，那么切线与弦 AB 的夹角等于弦 AB 在对面弓形中所对的圆周角（即 ∠ACB，其中 C 是圆上弦 AB 另一侧的任意一点）。

This theorem is often used when a diagram shows a tangent and a triangle inscribed in the circle. It can be proved by drawing the diameter through the point of contact and using the right angle from the tangent-radius theorem. In exams, be careful to identify the correct alternate segment.

当图形中出现一条切线和圆内接三角形时，经常用到这个定理。可以通过画过切点的直径并应用切线-半径定理的直角来证明。在考试中，要小心识别正确的另一侧弓形。

9. Equation of a Circle | 圆的方程

In the coordinate geometry part of the GCSE Edexcel syllabus, you need to recognise and use the equation of a circle centred at the origin:

x² + y² = r²

where (x, y) are coordinates of any point on the circle and r is the radius. For a circle with centre (a, b), the equation is

(x – a)² + (y – b)² = r²

在 GCSE Edexcel 大纲的坐标几何部分，你需要识别和使用以原点为圆心的圆的方程：

x² + y² = r²

其中 (x, y) 是圆上任意点的坐标，r 是半径。对于圆心在 (a, b) 的圆，方程为

(x – a)² + (y – b)² = r²

。

You can find the radius given the centre and a point on the circle by substituting into the equation. Conversely, you can determine whether a point lies inside, on, or outside the circle by comparing the left side value to r².

已知圆心和圆上一点，你可以通过代入方程求出半径。反过来，通过比较方程左边的值与 r²，可以判断一个点是在圆内、圆上还是圆外。

10. Worked Example & Exam Tips | 综合例题与应试技巧

Example: In a circle, O is the centre. Points A, B, and C lie on the circumference. Angle AOB = 124°. Find angle ACB.

例题：在圆中，O 为圆心，点 A、B、C 在圆周上。∠AOB = 124°。求 ∠ACB。

Solution: By the angle at the centre theorem, the angle at the circumference is half the angle at the centre. Thus ∠ACB = 124° ÷ 2 = 62°.

解答：根据圆心角定理，圆周角是圆心角的一半。因此 ∠ACB = 124° ÷ 2 = 62°。

Exam tips: Always write a brief reason for each step (e.g., ‘angle at centre = 2 × angle at circumference’). Look for isosceles triangles formed by radii, as base angles are equal. In multi-step problems, mark all known angles on the diagram. With tangents, immediately note the right angle. For cyclic quadrilaterals, highlight opposite angles. Practise past papers to become fluent in combining theorems.

应试技巧：每一步都写上简要的理由（如“圆心角 = 2 × 圆周角”）。寻找由半径构成的等腰三角形，因为底角相等。在多步问题中，将已知角度全部标在图上。遇到切线，立刻标记直角。对于圆内接四边形，突出显示对角。通过练习往年真题来熟练掌握定理的组合使用。

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GCSE Edexcel Maths: Circle Geometry Key Points | GCSE Edexcel 数学：圆周运动 考点精讲