IGCSE Maths: Circle Theorems | IGCSE 数学：圆定理

📚 IGCSE Maths: Circle Theorems | IGCSE 数学：圆定理

Circle theorems are a core part of IGCSE Mathematics. They describe the angle and length relationships that appear when lines, triangles and quadrilaterals interact with a circle. A solid understanding of these rules will help you solve both straightforward identification questions and multi-step problems in the Core and Extended papers.

圆定理是 IGCSE 数学的核心内容。它们描述了直线、三角形和四边形与圆相交时产生的角度与线段关系。扎实掌握这些规则将帮助你解决核心和扩展试卷中的直接识别题以及多步骤问题。

1. Circle Theorem Language | 圆定理术语

Before applying any theorem, you must be confident with the vocabulary. A chord is a straight line segment joining two points on the circumference. A tangent is a line that touches the circle at exactly one point. A secant is a line that cuts the circle at two points. An arc is part of the circumference, and a segment is the region between a chord and an arc.

在应用任何定理之前，你必须熟悉相关术语。弦是连接圆上两点的直线段。切线是与圆恰好相交于一点的直线。割线是与圆相交于两点的直线。弧是圆周的一部分，弓形是弦与弧之间的区域。

Radius: the distance from the centre to the circumference. 半径：从圆心到圆周的距离。
Diameter: a chord passing through the centre, equal to twice the radius. 直径：经过圆心的弦，长度等于半径的两倍。
Sector: the region between two radii and an arc. 扇形：两条半径与一段弧之间的区域。
Cyclic quadrilateral: a quadrilateral whose four vertices lie on the circle. 圆内接四边形：四个顶点都在圆上的四边形。

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

The angle at the centre of a circle is twice any angle at the circumference subtended by the same arc. If A, B and C lie on the circle and O is the centre, then the central angle ∠AOB equals 2 × ∠ACB. This theorem is often the first step in larger angle-chasing questions.

圆心角等于同弧所对圆周角的两倍。如果 A、B、C 在圆上，O 为圆心，那么圆心角 ∠AOB 等于 2 × ∠ACB。该定理通常是较大角度推理题的第一步。

∠AOB = 2 × ∠ACB

For example, if ∠ACB = 35°, then ∠AOB = 70°. If the central angle is 120°, the circumference angle on the same arc is 60°.

例如，如果 ∠ACB = 35°，那么 ∠AOB = 70°。如果圆心角是 120°，同弧上的圆周角就是 60°。

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

Angles in the same segment are equal. If two angles are subtended by the same chord on the same side of the chord, they have the same measure. In many diagrams, this appears as two angles standing on the same base chord but with their vertices at different points on the circumference.

同弦上的圆周角相等。如果两个角由同一条弦在同一侧所对，它们的度数相同。在许多图形中，这表现为两个角以同一条弦为底，但顶点位于圆周上的不同点。

∠APB = ∠AQB

This rule helps when you have multiple triangles drawn inside the same circle. Always check whether the angles really share the same chord and lie on the same side of it.

当你在同一个圆内画出多个三角形时，这条规则很有帮助。务必检查这些角是否真的共用同一条弦，并且位于该弦的同侧。

4. Angle in a Semicircle | 半圆上的圆周角

The angle in a semicircle is always a right angle, 90°. If AB is a diameter and C is any point on the circumference, then ∠ACB = 90°. This is a special case of the angle at the centre theorem, since the central angle for a diameter is 180°.

半圆上的圆周角始终是直角，即 90°。如果 AB 是直径，C 是圆周上的任意一点，那么 ∠ACB = 90°。这是圆心角定理的特例，因为直径所对的圆心角是 180°。

∠ACB = 90°

This theorem is particularly useful in coordinate geometry and Pythagoras problems. If you see a diameter in a circle, immediately look for a right-angled triangle.

该定理在坐标几何和勾股定理问题中特别有用。如果你在圆中看到直径，应立即寻找直角三角形。

5. Cyclic Quadrilaterals | 圆内接四边形

In a cyclic quadrilateral, the sum of each pair of opposite angles is 180°. If ABCD is a cyclic quadrilateral, then ∠A + ∠C = 180° and ∠B + ∠D = 180°. Another useful result is that an exterior angle of a cyclic quadrilateral equals the interior opposite angle.

在圆内接四边形中，每组对角之和为 180°。如果 ABCD 是圆内接四边形，那么 ∠A + ∠C = 180° 且 ∠B + ∠D = 180°。另一个有用的结论是，圆内接四边形的外角等于其内对角。

∠A + ∠C = 180° and ∠B + ∠D = 180°

This theorem often appears with parallel lines or with isosceles triangles, so do not forget that the opposite-angle rule only applies when all four vertices are confirmed to lie on the circle.

该定理常与平行线或等腰三角形一起出现，因此不要忘记，只有当四个顶点都确认在圆上时，对角互补规则才适用。

6. Tangent and Radius | 切线与半径

A tangent to a circle is perpendicular to the radius at the point of contact. This means that if OT is the radius and PT is the tangent, then ∠OTP = 90°. This fact is very common in questions involving Pythagoras’ theorem or trigonometry.

圆的切线垂直于经过切点的半径。这意味着如果 OT 是半径，PT 是切线，那么 ∠OTP = 90°。这一事实在涉及勾股定理或三角学的问题中非常常见。

∠OTP = 90°

When a tangent and a radius are shown, always mark the right angle immediately. It often unlocks the next step in finding missing lengths or angles.

当图中显示切线和半径时，务必立即标出直角。这通常会开启求未知长度或角度的下一步。

7. Tangents from an External Point | 圆外一点引切线

Two tangents drawn from the same external point to a circle are equal in length. If P is outside the circle and PA and PB are tangents touching the circle at A and B, then PA = PB. The line joining P to the centre bisects the angle between the tangents.

从圆外同一点引出的两条切线长度相等。如果 P 在圆外，PA 和 PB 是分别与圆相切于 A 和 B 的切线，那么 PA = PB。连接 P 与圆心的直线平分两条切线的夹角。

PA = PB

This theorem is often used to prove that two triangles are congruent, especially when combined with the tangent-radius perpendicularity rule.

该定理常用来证明两个三角形全等，特别是与切线-半径垂直规则结合使用时。

8. Alternate Segment Theorem | 弦切角定理

The angle between a tangent and a chord is equal to the angle in the alternate segment. If TAB is the angle between tangent TA and chord AB, and C is a point on the opposite arc, then ∠TAB = ∠ACB. This is one of the most frequently tested theorems in the Extended paper.

切线与弦之间的夹角等于弦所对的交替弓形内的圆周角。如果 TAB 是切线 TA 与弦 AB 之间的角，C 是另一侧弧上的点，那么 ∠TAB = ∠ACB。这是扩展试卷中最常考的定理之一。

∠TAB = ∠ACB

This theorem can be difficult to spot because the angle is formed by a tangent and a chord, not by two chords. Always look for a tangent touching the circle and a chord starting at the point of contact.

该定理较难识别，因为该角由切线和弦构成，而不是由两条弦构成。务必寻找与圆相切的切线以及从切点出发的弦。

9. Intersecting Chords Theorem | 相交弦定理

If two chords AB and CD intersect at a point P inside the circle, then the products of the lengths of the segments of each chord are equal. This means PA × PB = PC × PD. A similar product relationship also holds when two secants intersect outside the circle.

如果两条弦 AB 和 CD 在圆内点 P 相交，那么每条弦上两段长度的乘积相等。这意味着 PA × PB = PC × PD。当两条割线在圆外相交时，类似的乘积关系也成立。

PA × PB = PC × PD

This theorem is sometimes called the intersecting chord theorem. It is particularly useful when you are given some segment lengths and need to find an unknown length connecting two chords.

该定理有时称为相交弦定理。当你已知一些线段长度，并需要求出连接两条弦的未知长度时，它特别有用。

10. Problem Solving Strategy | 解题策略

When solving circle theorem problems, draw a large, clear diagram and mark all given angles, equal lengths and right angles. Identify which theorem links the known and unknown values. Work step by step and give a reason for every angle or length you calculate.

解圆定理题时，画出清晰的大图，并标出所有已知角、相等线段和直角。找出联系已知量和未知量的定理。逐步求解，并为你计算出的每个角或每条线段写出理由。

Find tangents and radii to mark right angles. 寻找切线和半径以标出直角。
Mark equal tangents and equal chords. 标记相等的切线和相等的弦。
Look for cyclic quadrilaterals and diameter-semicircle triangles. 寻找圆内接四边形和直径-半圆三角形。
Use the centre to connect central and circumference angles. 利用圆心连接圆心角和圆周角。
Do not forget to quote the theorem name or a short reason in the exam. 考试中不要忘记写出定理名称或简短理由。

11. Common Mistakes | 常见错误

Students often confuse the angle at the centre with the angle at the circumference, or assume that all angles in the same circle are equal. Another common error is treating any angle in a circle as 90°, but only angles in a semicircle have that property. Read the question carefully and check whether points lie on the circumference or at the centre.

学生常将圆心角和圆周角混淆，或认为同一个圆内的所有角都相等。另一个常见错误是把圆中的任意角都当作 90°，但只有半圆上的圆周角才具备这个性质。仔细读题，检查点是在圆周上还是在圆心。

Confusing the conditions of the alternate segment theorem. 混淆弦切角定理的条件。
Forgetting to give reasons for each angle. 忘记为每个角写出理由。
Misreading a tangent as a chord or a secant. 把切线误认为弦或割线。
Applying cyclic quadrilateral rules to non-cyclic quadrilaterals. 将圆内接四边形规则应用于非圆内接四边形。

12. Exam Practice Tips | 考试练习技巧

In IGCSE exams, circle theorem questions often appear as multi-step problems combining several rules. Practice with past paper questions under timed conditions. When checking your work, make sure each step follows logically from

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IGCSE Maths: Circle Theorems | IGCSE 数学：圆定理