最近有很多同學(xué)問我關(guān)于平面幾何的題目，那從今天開始做个，我們就來復(fù)習(xí)GCSE中相關(guān)的幾何知識，其中包括了圓的理論和復(fù)雜三角函數(shù)的應(yīng)用。由于三角函數(shù)中的正余弦定理本質(zhì)上是由圓內(nèi)的理論推導(dǎo)而來的腺晾，所以，今天我們首先總結(jié)關(guān)于圓的定理以及課本外的拓展知識辜贵。
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.
考點(diǎn)一悯蝉、圓的相關(guān)概念
1、圓的定義
在一個(gè)平面內(nèi)托慨，線段OA繞它固定的一個(gè)端點(diǎn)O旋轉(zhuǎn)一周鼻由，另一個(gè)端點(diǎn)A隨之旋轉(zhuǎn)所形成的圖形叫做圓，固定的端點(diǎn)O叫做圓心厚棵，線段OA叫做半徑蕉世。
2、圓的幾何表示
以點(diǎn)O為圓心的圓記作“⊙O”婆硬，讀作“圓O”?
考點(diǎn)二狠轻、弦、弧等與圓有關(guān)的定義
（1）弦：連接圓上任意兩點(diǎn)的線段叫做弦彬犯。（如圖中的AB）
（2）直徑：經(jīng)過圓心的弦叫做直徑向楼。（如圖中的CD）直徑等于半徑的2倍查吊。
（3）半圓：圓的任意一條直徑的兩個(gè)端點(diǎn)分圓成兩條弧，每一條弧都叫做半圓湖蜕。
（4）弧逻卖、優(yōu)弧、劣徽咽恪：圓上任意兩點(diǎn)間的部分叫做圓弧评也，簡稱弧。大于半圓的弧叫做優(yōu)桓曷场（多用三個(gè)字母表示）仇参；小于半圓的弧叫做劣弧（多用兩個(gè)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)
考點(diǎn)三婆殿、垂徑定理及其推論(重要)
垂徑定理：垂直于弦的直徑平分這條弦诈乒，并且平分弦所對的弧。
推論1：（1）平分弦（不是直徑）的直徑垂直于弦婆芦，并且平分弦所對的兩條弧怕磨。
（2）弦的垂直平分線經(jīng)過圓心，并且平分弦所對的兩條弧消约。
（3）平分弦所對的一條弧的直徑垂直平分弦肠鲫，并且平分弦所對的另一條弧。
*推論2：圓的兩條平行弦所夾的弧相等或粮。
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.
考點(diǎn)四导饲、圓的對稱性
1、圓的軸對稱性
圓是軸對稱圖形氯材，經(jīng)過圓心的每一條直線都是它的對稱軸渣锦。
2、圓的中心對稱性
圓是以圓心為對稱中心的中心對稱圖形氢哮。
考點(diǎn)五袋毙、弧、弦冗尤、弦心距听盖、圓心角之間的關(guān)系定理
1、圓心角
頂點(diǎn)在圓心的角叫做圓心角裂七。
2皆看、弦心距
從圓心到弦的距離叫做弦心距。
3碍讯、弧悬蔽、弦、弦心距捉兴、圓心角之間的關(guān)系定理
在同圓或等圓中蝎困，相等的圓心角所對的弧相等录语，所對的弦想等，所對的弦的弦心距相等禾乘。
推論：在同圓或等圓中澎埠，如果兩個(gè)圓的圓心角、兩條弧始藕、兩條弦或兩條弦的弦心距中有一組量相等蒲稳，那么它們所對應(yīng)的其余各組量都分別相等。
考點(diǎn)六伍派、圓周角定理及其推論
1江耀、圓周角
頂點(diǎn)在圓上，并且兩邊都和圓相交的角叫做圓周角诉植。
2祥国、圓周角定理（重要）
一條弧所對的圓周角等于它所對的圓心角的一半。
推論1：同弧或等弧所對的圓周角相等晾腔；同圓或等圓中舌稀，相等的圓周角所對的弧也相等。
推論2(△)：半圓（或直徑）所對的圓周角是直角灼擂；90°的圓周角所對的弦是直徑壁查。?
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.
