Circle Theorems | 圆定理

📚 Circle Theorems | 圆定理

Circle theorems are a set of powerful geometric rules that link angles, chords, tangents, and radii in a circle. For Edexcel IGCSE Mathematics, mastering these theorems is essential because they appear in both Paper 1 and Paper 2, often within multi-step proof or calculation questions.

圆定理是一系列强有力的几何规则，将圆中的角度、弦、切线和半径联系起来。对于 Edexcel IGCSE 数学，掌握这些定理至关重要，因为它们在试卷一和试卷二中都会出现，常融入多步证明或计算题中。

1. The Angle at the Centre Theorem | 圆心角定理

The angle at the centre of a circle is twice the angle at the circumference when both angles stand on the same arc (or chord). In the diagram below, if O is the centre and A, B, C are points on the circle, then ∠AOB = 2∠ACB.

当圆心角和圆周角对同一条弧（或弦）时，圆心角是圆周角的两倍。在下图中，若 O 是圆心，A、B、C 是圆上的点，则 ∠AOB = 2∠ACB。

If O is the centre, then ∠AOB = 2∠ACB

若 O 是圆心，则 ∠AOB = 2∠ACB

This theorem is often the first step in a proof, because it connects a central angle with an inscribed angle. For example, if ∠AOB = 70°, then any angle at the circumference subtending the same arc AB will be 35°.

这条定理常是证明的第一步，因为它将圆心角与圆周角联系起来。例如，若 ∠AOB = 70°，那么任何对同一条弧 AB 的圆周角都为 35°。

Be careful: the angle at the circumference must use the same chord or arc as the centre angle. If the point C lies on the opposite arc, the relationship becomes supplementary, not equal.

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