最近有很多同学问我关于平面几何的题目，那从今天开始，我们就来复习GCSE中相关的几何知识，其中包括了圆的理论和复杂三角函数的应用。由于三角函数中的正余弦定理本质上是由圆内的理论推导而来的，所以，今天我们首先总结关于圆的定理以及课本外的拓展知识。
Recently, many students have asked me about plane geometry. From today, we will review the relevant geometric knowledge in GCSE, including the theory of circles and the application of complex trigonometric functions. Since the sine and cosine theorem in trigonometric functions is essentially derived from the theory inside the circle, today we first summarize the theorems about circles and extended knowledge beyond the class.
考点一、圆的相关概念
1、圆的定义
在一个平面内，线段OA绕它固定的一个端点O旋转一周，另一个端点A随之旋转所形成的图形叫做圆，固定的端点O叫做圆心，线段OA叫做半径。
2、圆的几何表示
以点O为圆心的圆记作“⊙O”，读作“圆O”
考点二、弦、弧等与圆有关的定义
（1）弦：连接圆上任意两点的线段叫做弦。（如图中的AB）
（2）直径：经过圆心的弦叫做直径。（如图中的CD）直径等于半径的2倍。
（3）半圆：圆的任意一条直径的两个端点分圆成两条弧，每一条弧都叫做半圆。
（4）弧、优弧、劣弧：圆上任意两点间的部分叫做圆弧，简称弧。大于半圆的弧叫做优弧（多用三个字母表示）；小于半圆的弧叫做劣弧（多用两个字母表示）
(1) Chord: A line segment connecting any two points on a circle is called a chord. (As AB in the picture)
(2) Diameter: The chord passing through the center of the circle is called the diameter. (CD in the picture) The diameter is equal to 2 times the radius.
(3) Semicircle: The two endpoints of any diameter of a circle are divided into two arcs, and each arc is called a semicircle.
(4) Arc, superior arc, inferior arc: The part between any two points on a circle is called an arc, or arc for short. The arc larger than the semicircle is called the superior arc (represented by three letters); the arc smaller than the semicircle is called the inferior arc (represented by two letters)
考点三、垂径定理及其推论(重要)
垂径定理：垂直于弦的直径平分这条弦，并且平分弦所对的弧。
推论1：（1）平分弦（不是直径）的直径垂直于弦，并且平分弦所对的两条弧。
（2）弦的垂直平分线经过圆心，并且平分弦所对的两条弧。
（3）平分弦所对的一条弧的直径垂直平分弦，并且平分弦所对的另一条弧。
*推论2：圆的两条平行弦所夹的弧相等。
Test site 3. The vertical diameter theorem and its inference (important)
Vertical Diameter Theorem: The diameter perpendicular to the chord bisects the chord, and bisects the arc opposite the chord.
Corollary 1:
(1) The diameter that bisects a chord (not the diameter) is perpendicular to the chord, and bisects the two arcs that the chord opposes.
(2) The vertical bisector of the chord passes through the center of the circle and bisects the two arcs subtended by the chord.
(3) The diameter of one arc subtended by the chord bisects the chord vertically and bisects the other arc subtended by the chord.
*Corollary 2: The arcs contained by two parallel chords of a circle are equal.
考点四、圆的对称性
1、圆的轴对称性
圆是轴对称图形，经过圆心的每一条直线都是它的对称轴。
2、圆的中心对称性
圆是以圆心为对称中心的中心对称图形。
考点五、弧、弦、弦心距、圆心角之间的关系定理
1、圆心角
顶点在圆心的角叫做圆心角。
2、弦心距
从圆心到弦的距离叫做弦心距。
3、弧、弦、弦心距、圆心角之间的关系定理
在同圆或等圆中，相等的圆心角所对的弧相等，所对的弦想等，所对的弦的弦心距相等。
推论：在同圆或等圆中，如果两个圆的圆心角、两条弧、两条弦或两条弦的弦心距中有一组量相等，那么它们所对应的其余各组量都分别相等。
考点六、圆周角定理及其推论
1、圆周角
顶点在圆上，并且两边都和圆相交的角叫做圆周角。
2、圆周角定理（重要）
一条弧所对的圆周角等于它所对的圆心角的一半。
推论1：同弧或等弧所对的圆周角相等；同圆或等圆中，相等的圆周角所对的弧也相等。
推论2(△)：半圆（或直径）所对的圆周角是直角；90°的圆周角所对的弦是直径。
Test point 6. Circumferential angle theorem and its inference
1. Circumferential angle
An angle whose vertex is on a circle and intersects the circle on both sides is called a circumferential angle.
2. Circumferential Angle Theorem (important)
The circumferential angle subtended by an arc is half the central angle subtended by it.
Inference 1: The circumferential angles subtended by the same or equal arcs are equal; in the same or equal circles, the arcs subtended by equal circumferential angles are also equal.
Corollary 2 (△): The angle of the circumference subtended by the semicircle (or diameter) is a right angle; the chord subtended by the angle of the circumference of 90° is the diameter.
