📚 Circle Theorems for IGCSE | IGCSE 圆定理
Circle theorems are a key part of the IGCSE Mathematics syllabus. They describe the relationships between angles, chords, tangents, and radii in a circle. Many exam questions ask you to find missing angles or to prove a result, so you need to know each theorem precisely and to select the correct one quickly.
圆定理是 IGCSE 数学大纲中的核心内容。它们描述了圆中角、弦、切线和半径之间的关系。许多考试题目要求你求未知角或证明某个结论，因此你需要准确掌握每一条定理，并能快速选用正确的定理。
1. Angle at the Centre | 圆心角定理
AB subtends an angle at the centre O and an angle at a point P on the circumference, then ∠AOB = 2 × ∠APB.
圆心角等于同一条弧所对圆周角的两倍。如果劣弧 AB 在圆心 O 处所对的角为 ∠AOB，在圆周上一点 P 处所对的角为 ∠APB，那么 ∠AOB = 2 × ∠APB。
You can use this theorem in two directions: given the centre angle, halve it; given the circumference angle, double it. The rule applies when the angles are on the same side of the arc.
你可以双向使用这条定理：已知圆心角，除以 2；已知圆周角，乘以 2。该规则适用于同一弧同侧的角。
∠AOB = 2 × ∠APB
2. Angles in the Same Segment | 同弦上的圆周角
Angles in the same segment are equal. All angles subtended by the same chord on the same side of the chord, with their vertices on the circumference, have the same size.
同弦上的圆周角相等。由同一条弦在同一侧所对、顶点在圆周上的所有圆周角大小相同。
For example, if chord AB is drawn and points P and Q lie on the same arc AB, then ∠APB = ∠AQB.
例如，如果画出弦 AB，点 P 和 Q 位于同一条弧 AB 上，那么 ∠APB = ∠AQB。
This theorem is often combined with the angle at the centre theorem to find missing angles in segment diagrams.
该定理经常与圆心角定理结合使用，以求出弦段图形中的未知角。
∠APB = ∠AQB
3. Angle in a Semicircle | 半圆上的圆周角
e is always a right angle. If AB is a diameter, then for any point P on the circumference, ∠APB = 90°.
半圆上的圆周角始终是直角。如果 AB 是直径，那么对于圆周上的任意一点 P，∠APB = 90°。
This theorem is a special case of the angle at the centre theorem: the centre angle AOB is 180°, so the circumference angle is half of it.
这条定理是圆心角定理的特殊情形：圆心角 AOB 为 180°，因此圆周角是它的一半。
If AB is a diameter, ∠APB = 90°
4. Cyclic Quadrilaterals | 圆内接四边形
A cyclic quadrilateral is a four-sided shape whose vertices all lie on the circumference of a circle. The opposite angles of a cyclic quadrilateral add up to 180°.
圆内接四边形是指四个顶点都位于同一圆周上的四边形。圆内接四边形的对角之和为 180°。
If ABCD is cyclic, then ∠A + ∠C = 180° and ∠B + ∠D = 180°.
如果四边形 ABCD 是圆内接四边形，那么 ∠A + ∠C = 180°，∠B + ∠D = 180°。
The converse is also useful: if a pair of opposite angles in a quadrilateral add up to 180°, the quadrilateral is cyclic.
它的逆命题也很有用：如果一个四边形的一对对角之和为 180°，那么这个四边形是圆内接四边形。
∠A + ∠C = 180°, ∠B + ∠D = 180°
5. Radius and Tangent | 半径与切线
A tangent is a straight line that touches the circle at exactly one point. The radius drawn to the point of contact is perpendicular to the tangent.
切线是一条只与圆恰好有一个公共点的直线。连接切点与圆心的半径垂直于切线。
If OP is the radius and PT is the tangent at P, then ∠OPT = 90°.
如果 OP 是半径，PT 是过点 P 的切线，那么 ∠OPT = 90°。
This fact is very useful when a tangent is given and you need to form a right-angled triangle for Pythagoras or trigonometry.
当题目给出切线，而你需要构造直角三角形来使用勾股定理或三角函数时，这个结论非常有用。
∠OPT = 90°
6. Tangents from an External Point | 圆外一点引切线
in length. If T is an external point and A and B are the points of contact, then TA = TB.
从圆外一点引圆的两条切线长度相等。如果 T 是圆外一点，A 和 B 是切点，那么 TA = TB。
The line joining the external point to the centre bisects the angle between the two tangents. The equal tangent lengths also create an isosceles triangle.
连接圆外一点与圆心的直线平分两条切线之间的夹角。相等的切线长度也形成一个等腰三角形。
TA = TB
7. Alternate Segment Theorem | 弦切角定理
The alternate segment theorem says that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. If PT is a tangent at P and PQ is a chord, then the angle between PT and PQ equals the angle subtended by chord PQ in the opposite arc.
弦切角定理指出，过切点的切线与弦所夹的角等于该弦在另一条弧上所对的圆周角。如果 PT 是过点 P 的切线，PQ 是弦，那么 PT 与 PQ 所夹的角等于弦 PQ 在另一弧上所对的角。
This theorem often looks less obvious than the others. Label the tangent, the chord, and the angle in the alternate segment carefully before applying it.
这条定理通常看起来不如其他定理直观。在应用之前，要仔细标注切线、弦以及另一弧上的角。
