📚 Circle Theorems: Angle Properties of Circles | 圆定理：圆的角度性质
Circle theorems are a central part of the IGCSE Mathematics geometry syllabus. They describe the relationships between angles, chords, tangents, arcs, and the circle itself. Mastering these results allows you to solve multi-step angle problems efficiently, often without using trigonometry or coordinate geometry.
圆定理是 IGCSE 数学几何部分的核心内容。它们描述了角、弦、切线、弧与圆本身之间的关系。掌握这些结论可以帮助你高效解决多步角度问题，通常无需借助三角学或坐标几何。
1. Key Definitions and Notation | 关键定义与记号
A chord is a line segment whose endpoints both lie on the circle. A tangent is a line that touches the circle at exactly one point, called the point of contact. An arc is a portion of the circumference between two points. An angle is said to be subtended by an arc or chord at a point when the two rays forming the angle pass through the endpoints of that arc or chord.
弦是两端点都落在圆上的线段。切线是与圆恰好相交于一点的直线，该点称为切点。弧是圆周上两点之间的部分。当形成角的两条射线经过某弧或弦的端点时，称该角为这条弧或弦所对的角。
- Centre O: fixed point equidistant from all points on the circle.
- Chord AB: line segment with endpoints A and B on the circle.
- Arc AB: the curved part of the circle between A and B; the minor arc is shorter, the major arc is longer.
- Subtended angle ∠APB: angle at a point P formed by lines PA and PB.
圆心 O：到圆上所有点距离相等的定点。弦 AB：端点为 A 和 B 且都在圆上的线段。弧 AB：圆上 A 与 B 之间的弯曲部分；较短的为劣弧，较长的为优弧。对角 ∠APB：在点 P 处由直线 PA 和 PB 形成的角。
2. Theorem 1: Angle at the Centre | 定理一：圆心角
The angle at the centre of a circle is twice the angle at the circumference when both angles are subtended by the same arc. In symbols, if O is the centre and P is any point on the circumference on the same side of chord AB, then the central angle ∠AOB is twice the inscribed angle ∠APB.
圆心角等于同弧所对圆周角的两倍。用符号表示，如果 O 是圆心，P 是圆周上与弦 AB 同侧的任意点，那么圆心角 ∠AOB 是圆周角 ∠APB 的两倍。
∠AOB = 2 × ∠APB
ketch uses the fact that triangles OAP and OBP are isosceles because OA = OP = OB. Letting ∠OAP = ∠OPA = x and ∠OBP = ∠ − x − y), while the inscribed angle is 90° − x − y. This confirms the double relationship.
一个简短的证明思路是利用三角形 OAP 和 OBP 是等腰三角形，因为 OA = OP = OB。设 ∠OAP = ∠OPA = x，∠OBP = ∠OPB = y，则圆心角为 180° − 2x + 180° − 2y = 2(90° − x − y)，而圆周角为 90° − x − y。这证实了两倍关系。
3. Theorem 2: Angle in a Semicircle | 定理二：半圆角
If AB is a diameter of the circle, then for any point P lying on the circumference, the angle ∠APB is a right angle. This is a special case of the angle at the centre theorem: the central angle ∠AOB is 180°, so the inscribed angle is 90°.
如果 AB 是圆的直径，那么对于圆周上任意点 P，角 ∠APB 都是直角。这是圆心角定理的特殊情形：圆心角 ∠AOB 为 180°，因此圆周角为 90°。
If AB is a diameter → ∠APB = 90°
This result is extremely useful in problems where a diameter is given or must be identified. Look for a chord that passes through the centre; the angle at the circumference standing on that chord will always be a right angle. It also works in reverse: if an angle at the circumference is 90°, then the chord joining the endpoints of the two arms is a diameter.
这一结论在已知直径或需要识别直径的问题中非常有用。留意经过圆心的弦；以该弦为底边的圆周角始终是直角。反过来也成立：如果圆周上的一个角是 90°，那么连接该角两臂端点的弦就是直径。
4. Theorem 3: Angles in the Same Segment | 定理三：同弧上的圆周角
All angles subtended by the same chord on the same side of the chord are equal. That is, if P and Q are two points on the same arc AB, then the inscribed angles ∠APB and ∠AQB are equal.
同一条弦在同一侧所对的所有圆周角都相等。也就是说，如果 P 和 Q 是同一条弧 AB 上的两个点，那么圆周角 ∠APB 和 ∠AQB 相等。
∠APB
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com 更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图