Circle Theorems Explained | 圆定理详解

📚 Circle Theorems Explained | 圆定理详解

Circle theorems are a set of key results in IGCSE Mathematics that describe the relationships between angles, chords, tangents, and radii in a circle. Mastering these theorems is essential for solving geometry problems in both Paper 2 and Paper 4.

圆定理是一组在IGCSE数学中至关重要的结论，描述了圆中角、弦、切线和半径之间的关系。掌握这些定理对于解决试卷2和试卷4的几何题目至关重要。

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

In a circle, the angle formed at the centre by two radii (the central angle) is exactly twice the angle formed at the circumference by the same two points (the inscribed angle). If points A and B lie on the circle and point X is on the circumference (not within the arc AB), then ∠AOB = 2∠AXB.

在圆中，由两条半径在圆心处形成的角（圆心角）恰好是由同两点在圆周上形成的角（圆周角）的两倍。若点A和B在圆上，点X在圆周上（不在弧AB内），则 ∠AOB = 2∠AXB。

∠AOB = 2 × ∠AXB

Example: If ∠AXB = 50°, then the central angle ∠AOB cannot be seen directly in the diagram, but we know it must be 100°.

例如：若 ∠AXB = 50°，那么圆心角 ∠AOB 虽然不能直接量出，但我们知道它一定是 100°。

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

If two angles are inscribed in the same segment (i.e., they subtend the same chord), they are equal. For points A, B, C, D on the same circle, if ∠ACB and ∠ADB both subtend chord AB, then ∠ACB = ∠ADB.

如果两个圆周角对应同一条弧（即它们对着同一条弦），则这两个角相等。若点A、B、C、D在同一个圆上，且 ∠ACB 和 ∠ADB 都对着弦AB，那么 ∠ACB = ∠ADB。

∠ACB = ∠ADB

This theorem is especially useful when you need to find an unknown angle that shares a chord with a known angle.

当需要求解一个与已知角共用同一条弦的未知角时，这个定理特别有用。

3. The Angle in a Semicircle is 90° | 半圆内的圆周角是直角

If AB is a diameter of a circle, and C is any point on the circumference, then ∠ACB is always a right angle (90°). This is actually a special case of the first theorem, since the central angle ∠AOB is 180°.

如果AB是圆的直径，且C是圆周上任意一点，那么 ∠ACB 始终是直角（90°）。这实际上是第一个定理的特例，因为圆心角 ∠AOB 等于 180°。

∠ACB = 90°

This theorem is frequently tested in IGCSE, often combined with Pythagoras’ theorem to find lengths in right-angled triangles inscribed in a circle.

该定理在IGCSE考试中经常出现，常与勾股定理结合，用于求解圆内直角三角形中的边长。

4. Cyclic Quadrilaterals: Opposite Angles Sum to 180° | 圆内接四边形的对角互补

A quadrilateral is cyclic if all four vertices lie on a circle. In a cyclic quadrilateral, both pairs of opposite angles add up to 180°. For quadrilateral ABCD inscribed in a circle, ∠A + ∠C = 180° and ∠B + ∠D = 180°.

如果四边形的四个顶点都在同一个圆上，则称该四边形为圆内接四边形。在圆内接四边形中，两组对角之和均为 180°。若四边形ABCD内接于圆，则 ∠A + ∠C = 180°，且 ∠B + ∠D = 180°。

∠A + ∠C = 180°, ∠B + ∠D = 180°

Remember that this theorem works in the reverse direction too: if a quadrilateral has opposite angles summing to 180°, then it is cyclic.

注意该定理的逆命题也成立：如果四边形的对角之和为 180°，则它必定是圆内接四边形。

5. The Tangent-Radius Theorem | 切线与半径垂直定理

A tangent is a line that touches the circle at exactly one point. The radius drawn to the point of tangency is perpendicular to the tangent. If point T is on the circle and OT is the radius, then the tangent line at T is perpendicular to OT.

切线是与圆只有一个交点的直线。过切点的半径与切线互相垂直。若点T在圆上，OT为半径，则过点T的切线垂直于OT。

radius ⟂ tangent at the point of contact

For example, if ∠OTP is the angle between the radius and the tangent, then ∠OTP = 90°. This theorem is essential for proving other tangent-related results.

例如，若 ∠OTP 是半径与切线的夹角，则 ∠OTP = 90°。该定理是证明其他与切线相关结论的基础。

6. The Alternate Segment Theorem | 弦切角定理

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment (the angle subtended by the chord on the opposite arc). If tangent PT touches the circle at T, and chord TC is drawn, then the angle between PT and TC equals the angle subtended by TC at any point on the opposite side of the circle.

切线与其切点处的弦所夹的角，等于该弦所对的、位于切线另一侧的圆周角。若切线PT切圆于点T，弦TC已画出，则PT与TC之间的夹角，等于弦TC在圆对侧任意点处所对的圆周角。

∠PTC = ∠TAC

This theorem is often the key to solving problems that involve tangents and chords together.

在同时涉及切线和弦的问题中，这个定理常常是解题的关键。

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

If a line is drawn from the centre of a circle perpendicular to a chord, it will bisect the chord. Conversely, the perpendicular bisector of a chord passes through the centre of the circle. If OM is perpendicular to chord AB, then AM = MB.

从圆心引一条垂直于弦的线段，该线段会平分这条弦。反之，弦的垂直平分线必过圆心。若OM垂直于弦AB，则 AM = MB。

OM ⟂ AB ⇒ AM = MB

This theorem is very useful for calculating the radius of a circle when the chord length and its distance from the centre are known, by applying the Pythagorean theorem.

利用该定理，结合勾股定理，可在已知弦长和弦心距的情况下求出圆的半径。

8. Equal Tangents from an External Point | 从圆外一点引出的两条切线长度相等

From any point outside a circle, there are exactly two tangents to the circle, and their lengths are equal. If point P lies outside the circle and the tangents touch the circle at points A and B, then PA = PB.

从圆外任意一点，恰好可以作两条圆的切线，且这两条切线的长度相等。若点P在圆外，切点分别为A和B，则 PA = PB。

PA = PB

The radius-to-tangent theorem also tells us that OA is perpendicular to PA and OB is perpendicular to PB. This leads to congruent right triangles, which can be used to find other lengths and angles.

同时，半径与切线垂直定理告诉我们 OA ⊥ PA，OB ⊥ PB。由此可得到全等的直角三角形，进而求解其他边长和角度。

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Circle Theorems Explained | 圆定理详解