📚 Circle Theorems | 圆的定理
Circle theorems are fundamental results in IGCSE geometry that describe the relationships between angles, chords, tangents, and arcs within a circle. Mastering these theorems is essential for solving a wide range of exam questions and for proving geometric statements.
圆定理是IGCSE几何学中的基础结论，描述了圆内角、弦、切线和弧之间的内在联系。掌握这些定理对于解答各类考试题目以及完成几何证明至关重要。
1. Key Terms and Definitions | 关键术语与定义
Before exploring the theorems, it is essential to be familiar with the basic vocabulary of a circle. The centre is the fixed point equidistant from all points on the circumference. A radius is a line segment from the centre to any point on the circle, while a chord connects two points on the circumference without passing through the centre.
在探索定理之前，熟悉圆的基本术语至关重要。圆心是到圆周上所有点距离相等的固定点。半径是从圆心到圆上任意一点的线段，而弦是连接圆周上两点的线段，不经过圆心。
A diameter is a chord that passes through the centre, and it is the longest chord of the circle. A tangent is a straight line that touches the circle at exactly one point. An arc is a portion of the circumference, and a sector is the region bounded by two radii and an arc.
直径是经过圆心的弦，也是圆内最长的弦。切线是与圆恰好有一个交点的直线。弧是圆周的一部分，而扇形是由两条半径和一条弧围成的区域。
2. Theorem 1: Angle in a Semicircle | 定理1：半圆中的角
This theorem states that the angle subtended by a diameter at any point on the circumference is a right angle (90°). In other words, if AB is the diameter of a circle and C is any point on the circumference, then ∠ACB = 90°.
该定理指出：直径所对的圆周角为直角（90°）。换言之，若AB是圆的直径，C是圆周上任意一点，则∠ACB = 90°。
If AB is a diameter, then ∠ACB = 90°
若AB为直径，则∠ACB = 90°
This is one of the most frequently tested theorems in IGCSE papers. When you spot a triangle inscribed in a semicircle, you immediately know that the angle at the circumference is a right angle. This result is also known as Thales’ theorem and is the foundation for many longer proofs.
这是IGCSE考试中最常考查的定理之一。当你在半圆中看到一个内接三角形时，可以立即判断圆周上的角为直角。该结论也称为泰勒斯定理，是许多复杂证明的基础。
3. Theorem 2: Angle at the Centre | 定理2：圆心角定理
The angle at the centre of a circle is twice the angle at the circumference, provided that both angles subtend the same arc. If O is the centre and A, B, C are points on the circumference, then ∠AOB = 2 × ∠ACB.
圆心角等于同弧上圆周角的两倍。若O为圆心，A、B、C为圆周上的点，则∠AOB = 2 × ∠ACB。
∠AOB = 2 × ∠ACB (when both subtend arc AB)
∠AOB = 2 × ∠ACB（两者均对应弧AB时）
This theorem forms the basis for many other circle theorems. For example, since the diameter subtends a straight angle of 180° at the centre, the angle at the circumference must be 90°, which confirms the semicircle theorem from another perspective.
该定理是许多其他圆定理的基础。例如，由于直径在圆心处构成180°的平角，圆周角必为90°，这从另一个角度验证了半圆定理。
4. Theorem 3: Angles in the Same Segment | 定理3：同弧圆周角定理
Angles subtended by the same chord at different points on the circumference (on the same side of the chord) are equal. If points C and D both lie on the same side of chord AB, then ∠ACB = ∠ADB.
同一弦在圆周同侧所对的圆周角相等。若点C和D位于弦AB的同侧，则∠ACB = ∠ADB。
∠ACB = ∠ADB (when C and D are on the same side of chord AB)
∠ACB = ∠ADB（当C和D位于弦AB同侧时）
This theorem is extremely useful in geometric proofs. When you identify two angles subtending the same chord from the same side, you can immediately state that they are equal. This often serves as the key step in proving triangles similar or congruent.
该定理在几何证明中极为有用。当你识别出两个角从同侧对应同一条弦时，可以立即判断它们相等。这往往是证明三角形相似或全等的关键步骤。
5. Theorem 4: Cyclic Quadrilaterals | 定理4：圆内接四边形
A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the circumference of a circle. The opposite angles of a cyclic quadrilateral sum to 180°. If ABCD is cyclic, then ∠A + ∠C = 180° and ∠B + ∠D = 180°.
圆内接四边形是四个顶点都在同一圆圆周上的四边形。圆内接四边形的对角互补（之和为180°）。若ABCD为圆内接四边形，则∠A + ∠C = 180°，∠B + ∠D = 180°。
∠A + ∠C = 180°, ∠B + ∠D = 180
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