📚 IGCSE Maths G6 Teacher's Guide p.250: Circle Theorems | IGCSE 数学 G6 教师用书第250页：圆定理
This G6 Teacher's Guide page 250 revision article covers the circle theorems that appear most often in IGCSE Mathematics, both at Core and Extended levels. You will learn the precise angle facts, tangent results and cyclic quadrilateral rules needed to solve multi-step geometry problems with confidence.
这篇 G6 教师用书第250页复习文章涵盖了 IGCSE 数学核心与扩展阶段最常见的圆定理。你将学习精确的角的关系、切线结论以及圆内接四边形规则，从而自信地解决多步几何问题。
1. Basic Angle Language and Notation | 基本角的语言与记号
In circle geometry, an angle at the centre is formed by two radii, while an angle at the circumference is formed by two chords meeting at a point on the circle. Both angles are often subtended by the same arc AB, so we write ∠AOB for the centre angle and ∠ACB for the circumference angle.
在圆几何中，圆心角由两条半径构成，而圆周角由两条弦在圆上一点相交构成。两个角通常由同一条弧 AB 所对，因此我们用 ∠AOB 表示圆心角，用 ∠ACB 表示圆周角。
Always name the arc and points carefully: a chord AB divides the circle into a minor arc and a major arc. The same chord can produce different inscribed angles depending on which side of the chord the point C lies.
命名弧和点时要仔细：弦 AB 将圆分成劣弧和优弧。同一条弦根据点 C 位于弦的哪一侧，可能产生不同的内接角。
2. Theorem 1: Angle at the Centre | 定理一：圆心角定理
The first essential theorem states that the angle at the centre is twice any angle at the circumference subtended by the same arc. If O is the centre and A, B, C lie on the circle with C on the major arc AB, then:
第一个核心定理指出，圆心角等于同弧所对的任意圆周角的两倍。如果 O 是圆心，A、B、C 在圆上且 C 位于优弧 AB 上，那么：
∠AOB = 2 × ∠ACB
For example, if a minor arc AB subtends ∠ACB = 42° at the circumference, then the central angle ∠ n with isosceles triangles because OA and OB are both radii, so triangle AOB is isosceles.
例如，如果劣弧 AB 在圆周上对出 ∠ACB = 42°，那么圆心角 ∠AOB = 84°。该定理常与等腰三角形结合使用，因为 OA 和 OB 都是半径，所以三角形 AOB 是等腰三角形。
3. Theorem 2: Angle in a Semicircle | 定理二：半圆上的圆周角
When the chord AB is a diameter, the central angle ∠AOB becomes 180°, so any angle at the circumference subtended by the diameter is half of 180°:
当弦 AB 是直径时，圆心角 ∠AOB 变成 180°，因此直径所对的任意圆周角等于 180° 的一半：
∠ACB = 90°
This gives a very useful right-angled-triangle setup: if you are told AB is a diameter and C lies on the circle, triangle ACB is always right-angled at C. You can then apply Pythagoras' theorem or basic trigonometry.
这给出了一个非常有用的直角三角形结构：如果已知 AB 是直径且 C 在圆上，那么三角形 ACB 在 C 处一定是直角。然后你可以应用勾股定理或基本三角比。
Do not assume an angle is 90° just because it looks right; the diameter condition must be
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