G-7 Circle Theorems: Key Rules for IGCSE Geometry | G-7 圆的定理：IGCSE 几何核心规则

📚 G-7 Circle Theorems: Key Rules for IGCSE Geometry | G-7 圆的定理：IGCSE 几何核心规则

Circle theorems are a fundamental part of IGCSE Mathematics. They describe the relationships between angles, radii, chords, and tangents in a circle. Mastering these theorems is essential for solving geometry problems efficiently and accurately in your exams.

圆的定理是 IGCSE 数学的重要组成部分。它们描述了圆中角、半径、弦和切线之间的关系。掌握这些定理对于在考试中高效、准确地解决几何问题至关重要。

1. The Angle at the Centre is Twice the Angle at the Circumference | 圆心角是圆周角的两倍

This is one of the most important theorems. The angle subtended by an arc at the centre of the circle is exactly twice the angle subtended by the same arc at any point on the circumference.

这是最重要的定理之一。同一段弧所对的圆心角，恰好是这条弧在圆周上任意一点所对的圆周角的两倍。

∠AOB = 2 × ∠ACB

Here, O is the centre of the circle, and C is any point on the circumference. This theorem applies when A, B, and C all lie on the circle and O is the centre.

这里，O 是圆心，C 是圆周上的任意一点。当 A、B、C 都在圆上且 O 为圆心时，该定理成立。

For example, if ∠AOB = 80°, then ∠ACB = 40°. This relationship is the foundation for many other circle theorems and is frequently tested in IGCSE papers.

例如，如果 ∠AOB = 80°，则 ∠ACB = 40°。这一关系是许多其他圆的定理的基础，在 IGCSE 试卷中经常出现。

2. Angles in the Same Segment are Equal | 同弧上的圆周角相等

Angles subtended by the same chord at different points on the circumference are equal, provided they are in the same segment of the circle.

同一弦在圆周上不同点所对的角，只要位于同一圆内同一弓形内，则这些角相等。

∠APB = ∠AQB

In this case, points P and Q lie on the same arc AB. The angles ∠APB and ∠AQB are both subtended by chord AB and therefore have equal measures.

在这种情况下，点 P 和 Q 位于同一段弧 AB 上。∠APB 和 ∠AQB 均由弦 AB 所对，因此它们的角度相等。

This theorem is especially useful when solving problems that involve multiple points on the circumference. Always check whether two angles share the same chord before applying this rule.

这个定理在解决涉及圆周上多个点的题目时特别有用。应用此规则前，务必检查两个角是否对应同一条弦。

3. The Angle in a Semicircle is a Right Angle | 半圆中的圆周角是直角

If a triangle is inscribed in a circle such that one of its sides is the diameter, then the angle opposite that diameter is always 90°.

如果一个三角形内接于圆，且其中一条边是直径，那么该直径所对的角始终为 90°。

∠ACB = 90°, where AB is the diameter

Here, AB is the diameter of the circle, and C is any point on the circumference. The angle at C is always a right angle, regardless of where C is located on the semicircle.

这里，AB 是圆的直径，C 是圆周上的任意一点。无论 C 位于半圆上的哪个位置，C 处的角始终是直角。

This theorem is often used in combination with Pythagoras’ theorem or trigonometry to find missing side lengths in right-angled triangles.

该定理常与勾股定理或三角函数结合使用，以求解直角三角形中未知边的长度。

4. Opposite Angles of a Cyclic Quadrilateral Sum to 180° | 圆内接四边形的对角和为 180°

A cyclic quadrilateral is a four-sided shape whose vertices all lie on the circumference of a circle. The sum of each pair of opposite angles in such a quadrilateral is always 180°.

圆内接四边形是指四个顶点都在同一圆周上的四边形。这种四边形中每一对对角的和始终为 180°。

∠A + ∠C = 180°
∠B + ∠D = 180°

This theorem is directly derived from the angle-at-the-centre theorem. Since the full angle around the centre is 360°, the opposite angles must divide this into two pairs summing to 180° each.

该定理由圆心角定理直接推导而来。由于圆心周围的完整角度为 360°，对角必须将其分成两组，每组和为 180°。

This is one of the most common theorems in IGCSE exam questions involving quadrilaterals. Be sure to identify cyclic quadrilaterals correctly before applying this rule.

这是 IGCSE 考试中涉及四边形的常见定理之一。在应用此规则前，务必正确识别圆内接四边形。

5. The Tangent and Radius Meet at a Right Angle | 切线与半径垂直

A tangent to a circle is a straight line that touches the circle at exactly one point. The radius drawn to that point of contact is always perpendicular to the tangent.

圆的切线是与圆只有一个交点的直线。连接圆心与该交点的半径始终垂直于切线。

OA ⊥ Tangent at point A

If O is the centre and A is the point of tangency, then the radius OA forms a 90° angle with the tangent line at A.

如果 O 是圆心，A 是切点，那么半径 OA 与 A 点处的切线形成 90° 角。

This theorem is frequently used in problems involving circle equations or tangent line calculations. It also helps in constructing right-angled triangles for further analysis.

该定理常用于涉及圆的方程或切线计算的题目中。它还有助于构造直角三角形以进行进一步分析。

6. Tangents from an External Point are Equal | 从圆外一点引出的两条切线等长

If two tangents are drawn to a circle from the same external point, the lengths of these two tangent segments are equal.

如果从同一个圆外一点向圆引两条切线，那么这两条切线段长度相等。

PA = PB

Here, P is a point outside the circle, and PA and PB are tangents touching the circle at points A and B, respectively. The distances PA and PB are always equal.

这里，P 是圆外一点，PA 和 PB 分别切圆于 A 点和 B 点。距离 PA 与 PB 始终相等。

Combined with the previous theorem, this gives rise to two congruent right-angled triangles, which can be used to solve for unknown lengths or angles.

结合前一个定理，可以形成两个全等的直角三角形，可用于求解未知边长或角度。

7. The Alternate Segment Theorem | 弦切角定理（切线与弦所夹的角）

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.

切线与过切点的弦所夹的角，等于该弦所对的圆周角（即圆中另一弓形内的角）。

∠TAB = ∠ACB

In this setup, TA is a tangent at point A, and AB is a chord. The angle between the tangent and the chord (∠TAB) equals the angle subtended by the chord at any point C on the opposite side of the circle.

在此图中，TA 是 A 点处的切线，AB 是弦。切线与弦之间的角（∠TAB）等于弦 AB 在圆对侧任意点 C 所对的圆周角。

This theorem is less intuitive but appears frequently in higher-level IGCSE questions. Practice identifying alternate segments to apply it confidently.

该定理不太直观，但在 IGCSE 高难度题目中经常出现。多加练习识别不同弓形，即可熟练运用。

8. The Perpendicular from the Centre to a Chord Bisects the Chord | 圆心到弦的垂线平分弦

A perpendicular drawn from the centre of a circle to a chord divides the chord into two equal parts. The perpendicular line also bisects the angle subtended by the chord at the centre.

从圆心向弦作垂线，该垂线将弦分成两段相等的部分。这条垂线还平分该弦在圆心处所对的角。

AM = MB, given OM ⊥ AB

Here, O is the centre, AB is the chord, and M is the midpoint of the chord. The line OM is perpendicular to AB, and AM = MB.

这里，O 是圆心，AB 是弦，M 是弦的中点。直线 OM 垂直于 AB，且 AM = MB。

In many problems, the perpendicular distance from the centre to a chord can be found using the Pythagorean theorem. This theorem is especially useful when working with chord lengths and circle geometry.

在许多题目中，可以利用勾股定理求从圆心到弦的垂直距离。该定理在涉及弦长和圆几何的问题中特别有用。

9. Solving Problems with Multiple Circle Theorems | 综合运用多个圆的定理解题

In IGCSE exams, you will often encounter questions that require applying more than one circle theorem at a time. A good strategy is to mark all known angles and identify which theorems apply before writing any equations.

在 IGCSE 考试中，你经常会遇到需要同时应用多个圆的定理的题目。一个好的策略是：先标出所有已知角，判断哪些定理适用，然后再列方程。

For example, if you know that AB is the diameter and you also know a tangent length, you might combine the semicircle right-angle theorem with the equal tangents theorem to find missing values.

例如，若已知 AB 是直径，同时知道一条切线的长度，你可以将半圆周角定理与等切线定理结合，求未知量。

Always draw a clear diagram and label every point. Write down the theorem name you use for each step in your working — examiners reward clearly reasoned solutions.

务必画出清晰的图形并标注每个点。在解题每一步旁边写下你所用的定理名称——考官会对逻辑清晰的解答给予肯定。

10. Common Mistakes to Avoid | 常见错误

Many students lose marks in circle theorem questions due to a few avoidable errors. Here are the most common ones to watch out for.

许多学生在圆的定理题目中丢分，往往是源于一些可以避免的错误。以下是常见的陷阱。

Using the angle-at-the-centre theorem when the vertex is not at the centre — the theorem only applies when the vertex is the exact centre of the circle.

在顶点不是圆心时使用圆心角定理——该定理仅在顶点恰好是圆心时才成立。
Assuming any quadrilateral is cyclic — all four vertices must lie on the circumference for cyclic quadrilateral rules to apply.

假设任意四边形都是圆内接四边形——只有当四个顶点都在圆周上时，圆内接四边形定理才适用。
Confusing tangents with chords — a tangent touches the circle at exactly one point and never crosses the interior.

混淆切线与弦——切线与圆只有一个交点，且永不穿过圆内。
Forgetting to justify the alternate segment theorem — always name the chord and the tangent when applying it.

在使用弦切角定理时忘记说明理由——应用时务必指出对应的弦和切线。

Careful reasoning and correct theorem selection will help you avoid these pitfalls.

仔细推理并正确选择定理，能帮助你避免这些陷阱。

11. Worked Example | 例题精讲

Let us work through a typical exam-style question step by step. The diagram features a circle with centre O, a tangent TA at point A, and points B and C on the circumference. Given that ∠TAB = 50° and ∠AOB = 100°, find ∠ABC.

让我们逐步分析一道典型考试风格题目。图中有一个圆心为 O 的圆，切线 TA 切于 A 点，B 和 C 是圆周上的点。已知 ∠TAB = 50° 和 ∠AOB = 100°，求 ∠ABC。

Step 1: Since TA is a tangent and OA is a radius, ∠OAT = 90°. Therefore, ∠OAB = 90° − 50° = 40°.

第一步：由于 TA 是切线，OA 是半径，所以 ∠OAT = 90°。因此，∠OAB = 90° − 50° = 40°。

Step 2: In triangle OAB, OA = OB (both are radii), so the triangle is isosceles. Thus, ∠OBA = ∠OAB = 40°. The remaining angle ∠AOB = 100°, which is consistent with the given value.

第二步：在三角形 OAB 中，OA = OB（二者均为半径），所以该三角形是等腰三角形。因此，∠OBA = ∠OAB = 40°。剩余角 ∠AOB = 100°，与题目给定值一致。

Step 3: The angle subtended by arc AB at the centre is 100°, so the angle at the circumference subtended by the same arc is 50°. Hence ∠ACB = 50°.

第三步：弧 AB 所对的圆心角为 100°，因此同弧所对的圆周角为 50°。故 ∠ACB = 50°。

Step 4: Since ∠ABC is the angle between BA and BC, we use the cyclic triangle ABC to note that ∠ABC = 180° − ∠ACB − ∠BAC. But ∠BAC = ∠TAB = 50° by the alternate segment theorem. Therefore ∠ABC = 180° − 50° − 50° = 80°.

第四步：因为 ∠ABC 是 BA 与 BC 之间的角，利用三角形 ABC，得 ∠ABC = 180° − ∠ACB − ∠BAC。由弦切角定理，∠BAC = ∠TAB = 50°。因此 ∠ABC = 180° − 50° − 50° = 80°。

Thus, the final answer is 80°. This example demonstrates the power of combining multiple circle theorems in one solution.

因此，最终答案为 80°。这个例子展示了在同一个解答中综合运用多个圆的定理的功效。

12. Summary and Final Tips | 总结与温馨提示

Circle theorems are not just rules to memorise — they are tools for logical reasoning. To succeed in IGCSE Mathematics, practise drawing clear diagrams, labelling all given information, and citing the correct theorem at each step.

圆的定理不仅是需要记忆的规则，更是逻辑推理的工具。要在 IGCSE 数学中取得好成绩，请练习绘制清晰图形、标注所有已知信息，并在每一步引用正确的定理。

Learn the names and statements of all six core theorems.

熟记全部六条核心定理的名称与表述。
Practise past paper questions in a timed setting to build speed and confidence.

限时练习历年试题，以提高速度和信心。
Always include a reason in your working — for example, “angle in a semicircle” or “alternate segment theorem”.

在解题过程中写明理由——例如 “半圆中的角” 或 “弦切角定理”。
Review your diagram from the examiner’s perspective — does each angle make sense?

从考官视角审视你的图形——每个角是否合理？

With regular practice, you will find that circle theorem problems become intuitive and even enjoyable. Good luck with your revision!

通过规律性练习，你会发现圆的定理题目会变得非常直观，甚至充满趣味。祝你复习顺利！

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G-7 Circle Theorems: Key Rules for IGCSE Geometry | G-7 圆的定理：IGCSE 几何核心规则