📚 Tangent and Chord Properties | 切线性质与弦性质
In A-Level Edexcel Maths, circle geometry plays a significant role, particularly in the pure mathematics component. Understanding the properties of tangents and chords allows you to solve problems involving angles, lengths, and coordinate geometry. This article covers the key theorems, their proofs, and common exam applications to help you master this topic.
在A-Level Edexcel数学中，圆几何在纯数学部分占有重要地位。掌握切线和弦的性质，能够解决涉及角度、长度和坐标几何的问题。本文涵盖关键定理、其证明以及常见的考试应用，帮助你彻底掌握这一主题。
1. Basic Definitions of Tangents and Chords | 切线、弦的基本定义
A tangent is a line that touches the circle at exactly one point, called the point of tangency. The tangent never enters the interior of the circle. A chord is a segment whose endpoints both lie on the circle; the longest chord is the diameter. A secant is a line that intersects the circle at two points. These elements are the building blocks of circle theorems.
切线是一条与圆恰好相切于一点的直线，该点称为切点。切线不会进入圆的内部。弦是一条两端都在圆上的线段；最长的弦是直径。割线是一条与圆相交于两点的直线。这些元素是圆定理的基础构件。
2. Tangent-Radius Perpendicularity | 切线与半径垂直
One of the most fundamental properties is that the tangent at any point on a circle is perpendicular to the radius drawn to that point. So, if a line touches the circle at point A, and O is the centre, then OA is perpendicular to the tangent line. This theorem is often used to find angles or to prove that a line is a tangent.
Proof sketch: Suppose the tangent is not perpendicular; then another line from O to the tangent would be shorter than the radius, causing the tangent to cut the circle, contradicting the definition. Therefore, it must be perpendicular.
最基本的性质之一是：圆上任意一点的切线与过该点的半径垂直。因此，如果一条直线与圆相切于点A，圆心为O，则OA垂直于该切线。这个定理常用于求角度或证明一条直线是切线。
证明概要：假设切线不垂直，那么从圆心到切线可以引一条比半径更短的线段，这将使切线与圆相交，与切线定义矛盾。因此，切线必须与半径垂直。
3. Two Tangents from an External Point | 从外部点引出的两条切线
ternal point to the centre of the circle bisects the angle between the two tangents. For example, if P is an external point and PA, PB are tangents touching the circle at A and B, then PA = PB and ∠APO = ∠BPO, where O is the centre.
s OAP and OBP are congruent (right-angled at the tangency points, OA=OB radii, OP common).
如果从圆外一点引两条切线到圆上，则这两条切线段的长度相等。此外，连接外部点与圆心的直线平分这两条切线所夹的角。例如，若P为圆外一点，PA、PB是切于点A和B的切线，则PA = PB，且∠APO = ∠BPO（O为圆心）。
可通过证明三角形OAP和OBP全等来证明（切点处为直角，半径OA=OB，公共边OP）。
4. Perpendicular Bisector of a Chord | 弦的垂直平分线
There is an important relationship between the centre of a circle and a chord: the line drawn from the centre of a circle perpendicular to a chord bisects the chord. Conversely, the line joining the centre to the midpoint of a chord is perpendicular to the chord. Thus, the perpendicular bisector of any chord passes through the centre of the circle. This property is essential for finding the centre or solving problems involving chord lengths.
If chord AB has midpoint M, then OM ⊥ AB and AM = MB.
圆心与弦之间存在一个重要关系：从圆心作垂直于弦的直线平分该弦。反之，连接圆心与弦中点的直线垂直于该弦。因此，任何弦的垂直平分线都经过圆心。这个性质对于求圆心或解决涉及弦长的问题至关重要。
如果弦AB的中点为M，则OM ⊥ AB且AM = MB。
5. Intersecting Chords Theorem | 相交弦定理
When two chords intersect inside a circle, the products of the lengths of the segments created on each chord are equal
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