📚 IGCSE Maths Circle Theorems: Angle Rules Made Easy | IGCSE 数学圆定理：轻松掌握角度规则
Circle theorems tell you how angles behave when points, chords, tangents and radii are drawn on a circle. In IGCSE Mathematics, these rules appear frequently in both non-calculator and calculator papers.
圆定理描述了点、弦、切线和半径画在圆上时角的变化规律。在 IGCSE 数学中，这些规则经常出现在非计算器和计算器试卷中。
1. Angle at the Centre | 圆心角定理
If an arc AB subtends angle ∠AOB at the centre and angle ∠APB at the circumference, then the central angle is twice the circumference angle.
如果弧 AB 在圆心处所对的是 ∠AOB，在圆周上同侧点 P 处所对的是 ∠APB，那么圆心角等于圆周角的两倍。
∠AOB = 2 × ∠APB
You can remember this as 'the centre angle is double the edge angle'. Always check that both angles are subtended by the same arc AB.
你可以记为“圆心角是圆周角的两倍”。使用该规则时一定要确认两个角都由同一条弧 AB 所对。
2. Angle in a Semicircle | 半圆角定理
An angle drawn inside a semicircle with its arms ending at the diameter endpoints is always 90°.
以直径为斜边、顶点在圆上的圆周角始终等于 90°。
If AB is a diameter, then ∠APB = 90°
This is a special case of Section 1 because the central angle is 180°, so the angle at the circumference is half of 180°. This theorem is often used with Pythagoras' theorem because the right angle creates a right-angled triangle.
这是第1节定理的特殊情况：圆心角为 180°，所以圆周角为 180° 的一半。该定理常与勾股定理一起使用，因为直角可以构造直角三角形。
3. Angles in the Same Segment | 同弓形角定理
Angles subtended by the same chord AB and lying in the same segment are equal.
由同一条弦 AB 所对、且位于同一弓形中的圆周角相等。
∠APB = ∠AQB
P and Q must be on the same side of chord AB. Do not use this rule if the points lie on opposite sides of the chord; that is a cyclic quadrilateral situation.
点 P 和 Q 必须位于弦 AB 的同一侧。如果两个点位于弦的两侧，则不能使用该规则；那属于圆内接四边形的情形。
4. Cyclic Quadrilateral | 圆内接四边形对角互补
If a quadrilateral has all four vertices on the circumference, it is cyclic. Opposite angles of a cyclic quadrilateral sum to 180°.
如果四边形的四个顶点都在圆周上，它就是圆内接四边形。圆内接四边形的对角之和为 180°。
∠A + ∠C = 180° and ∠B + ∠D = 180°
You can find one angle from the opposite angle by subtracting from 180°. You can also use this rule to prove a quadrilateral is cyclic if a pair of opposite angles add to 180°.
你可以用 180° 减去一个对角来求出另一个对角。若一组对角之和为 180°，也可反推出该四边形是圆内接四边形。
5. Tangent and Radius | 切线与半径垂直
A tangent to a circle meets the radius at the point of contact at a right angle.
圆的切线与经过切点的半径垂直。
OP ⟂ tangent at P, so ∠OPT = 90°
This is one of the most useful facts for finding unknown lengths. If you know two lengths in the right-angled triangle OTP, use Pythagoras' theorem to find the third.
这是求未知长度时最有用的性质之一。如果在直角三角形 OTP 中已知两条边，就可以用勾股定理求第三边。
6. Two Tangents from an External Point | 圆外一点引两切线
From an external point T, the two tangent segments TA and TB drawn to the circle are equal in length.
从圆外一点 T 引圆的两条切线 TA 和 TB，切线长相等。
TA = TB
The line joining T to the centre O bisects the angle between the tangents. Triangles OTA and OTB are congruent, so the angles at T and O are also equal in pairs.
连接 T 与圆心 O 的直线平分两切线之间的夹角。三角形 OTA 与 OTB 全等，因此 T 处和 O 处的角也分别相等。
7. Alternate Segment Theorem | 弦切角定理
The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.
过切点的弦与切线所夹的角，等于该弦所对的另一侧圆周角。
∠TBA = ∠BCA
Here AB is the chord, BT is the tangent, and C lies in the alternate segment. This theorem is useful when a tangent and a chord are given, and you need an angle on the other side of the circle.
图中 AB 为弦，BT 为切线，C 位于另一弓形中。当题目给出切线和弦，并要求求圆另一侧的角时，这个定理非常有用。
8. Perpendicular from Centre to Chord | 圆心到弦的垂线平分弦
If a line from the centre of a circle is perpendicular to a chord, it bisects that chord.
如果从圆心作一条弦的垂线，那么这条垂线平分该弦。
If OM ⟂ AB, then AM = MB
The converse is also true: a line joining the centre to the midpoint of a chord is perpendicular to the chord. This theorem can also be used with Pythagoras to find the radius or chord length.
逆命题也成立：连接圆心与弦中点的直线垂直于该弦。该定理也可以与勾股定理结合来求半径或弦长。
9. Radii Create Isosceles Triangles | 半径构成等腰三角形
Any triangle formed by two radii and a chord is isosceles because both radii are equal.
由两条半径和一条弦构成的三角形是等腰三角形，因为两条半径长度相等。
OA = OB, so ∠OAB = ∠OBA
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