Circle Theorems: IGCSE Geometry Core Skills | 圆定理：IGCSE几何核心技能

📚 Circle Theorems: IGCSE Geometry Core Skills | 圆定理：IGCSE几何核心技能

Circle theorems are a central part of IGCSE geometry. They describe relationships between angles, chords, tangents and radii in a circle. Mastering them allows you to solve angle problems efficiently and to write clear geometric proofs.

圆定理是 IGCSE 几何的核心内容。它们描述了圆中角、弦、切线和半径之间的关系。掌握这些定理可以帮助你高效解决角度问题，并写出清晰的几何证明。

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1. Reading the Diagram: Recognising the Circle | 读懂图形：识别圆

Before applying any circle theorem, identify the key elements: centre O, radii, chords, tangents, arcs, and points on the circumference. A diagram often hides the theorem in its construction.

在应用任何圆定理之前，先识别关键元素：圆心 O、半径、弦、切线、弧以及圆周上的点。图形常常把定理隐藏在它的构造中。

Label all given angles and lengths. If the centre is not given, do not assume a point is the centre unless the question says so.

标出所有已知角度和边长。如果没有给出圆心，不要假设某个点是圆心，除非题目明确说明。

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

If O is the centre of the circle, the angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the circumference.

如果 O 是圆心，那么同一段弧所对的圆心角等于该弧所对任意圆周角的两倍。

In symbols: ∠AOB = 2 × ∠ACB, where A and B are points on the circle and C is another point on the circumference.

符号表示：∠AOB = 2 × ∠ACB，其中 A、B 是圆上的点，C 是圆周上的另一点。

This rule must be applied only when the angle at the centre and the angle at the circumference are subtended by the same arc or chord.

该规则仅在圆心角和圆周角同对同一段弧或同一条弦时适用。

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

Angles in the same segment 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, they have the same size.

同弦上的圆周角相等。这意味着如果两个角由同一条弦或同一段弧所对，并且它们的顶点位于弦的同一侧，那么它们大小相等。

e, if points A, B, C and D lie on a circle with C and D on the same side of chord AB, then ∠ACB = ∠ADB.

例如，若 A、B、C、D 在同一个圆上，且 C、D 在弦 AB 的同一侧，则 ∠ACB = ∠ADB。

Do not apply this rule to angles on opposite sides of the chord; those angles are supplementary, not equal.

不要将这个规则应用于弦两侧的角；弦两侧的圆周角互补，而不是相等。

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4. Angle in a Semicircle | 半圆上的圆周角

angle, 90°. If AB is a diameter of the circle, then for any point C on the circumference, ∠ACB = 90°.

半圆上的圆周角始终是直角，即 90°。如果 AB 是圆的直径，那么对于圆周上的任意点 C，∠ACB = 90°。

This theorem is a special case of the centre angle rule: the central angle AOB is 180°, so the angle at the circumference is half of 180°, which is 90°.

该定理其实是圆心角定理的特例：圆心角 AOB 为 180°，所以圆周角是 180° 的一半，即 90°。

Always look for a diameter in the diagram, because it often gives a hidden right angle that unlocks the rest of the problem.

始终在图形中寻找直径，因为直径常常隐藏一个直角，而这个直角能解开其余问题。

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5. Cyclic Quadrilaterals | 圆内接四边形

A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the circumference of a circle. In any cyclic quadrilateral, opposite angles add up to 180°.

圆内接四边形是指四个顶点都在同一个圆上的四边形。任意圆内接四边形的对角互补，即对角之和为 180°。

If ABCD is cyclic, then ∠A + ∠C = 180° and ∠B + ∠D = 180°.

如果 ABCD 是圆内接四边形，则 ∠A + ∠C = 180°，∠B + ∠D = 180°。

Also, the exterior angle at a vertex equals the interior opposite angle. For example, if side AD is extended, the exterior angle at D equals ∠B.

此外，一个顶点处的外角等于内对角。例如，如果延长边 AD，那么 D 处的外角等于 ∠B。

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6. Tangent and Radius | 切线与半径

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

圆的切线是与圆只有一个公共点的直线。圆上一点的切线垂直于过该点的半径。

If OT is the radius to the point of contact T and line L is the tangent at T, then OT ⟂ L, so ∠OTL = 90°.

如果 OT 是

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