📚 AS Mathematics: Coordinate Geometry Revision Guide | AS 数学：坐标几何考点精讲
Coordinate geometry, also known as analytic geometry, bridges algebra and geometry by representing geometric figures using coordinates. Mastering its principles is essential for success in AS Mathematics. This guide covers key formulas and problem-solving strategies.
坐标几何（解析几何）通过坐标表示几何图形，将代数与几何联系起来。掌握其核心原理是 AS 数学取得高分的关键。本指南涵盖重要公式和解题技巧。
1. Cartesian Coordinates and Basic Formulas | 笛卡尔坐标系与基本公式
In the Cartesian plane, any point is identified by an ordered pair (x, y). The distance between two points A(x₁, y₁) and B(x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²].
在笛卡尔平面上，任意一点由有序对 (x, y) 表示。两点 A(x₁, y₁) 与 B(x₂, y₂) 的距离为 d = √[(x₂ − x₁)² + (y₂ − y₁)²]。
The midpoint M of segment AB has coordinates M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ).
线段 AB 的中点 M 坐标为 M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )。
These formulas are foundational and appear frequently in problems involving triangles, quadrilaterals, and circles.
这些公式是基础，经常出现在涉及三角形、四边形和圆的问题中。
2. Gradient of a Straight Line | 直线的斜率
The gradient (slope) m of the line passing through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁) / (x₂ − x₁). This measures the steepness and direction of the line.
经过 (x₁, y₁) 与 (x₂, y₂) 的直线斜率 m 为 m = (y₂ − y₁) / (x₂ − x₁)，它衡量了直线的倾斜程度和方向。
A positive gradient indicates an increasing line, a negative gradient a decreasing line, zero gradient a horizontal line (y = constant), and undefined gradient for a vertical line (x = constant).
斜率为正表示直线上升，斜率为负下降，斜率为零表示水平线 (y = 常数)，斜率无定义表示垂直线 (x = 常数)。
The angle θ the line makes with the positive x-axis satisfies tanθ = m.
直线与 x 轴正方向的夹角 θ 满足 tanθ = m。
3. Equations of a Straight Line | 直线方程的形式
The most common forms are:
常见形式有：
- Point-gradient form: y − y₁ = m(x − x₁), where (x₁, y₁) is a point on the line and m is the gradient.
- 点斜式: y − y₁ = m(x − x₁)，其中 (x₁, y₁) 是直线上一点，m 是斜率。
- Slope-intercept form: y = mx + c, where c is the y-intercept (the point where the line crosses the y-axis).
- 斜截式: y = mx + c，c 是 y 轴截距（直线与 y 轴交点的纵坐标）。
- General form: Ax + By + C = 0, where A, B, C are integers and A ≥ 0 if possible. The gradient can be found from −A/B.
- 一般式: Ax + By + C = 0，其中 A, B, C 为整数，通常使 A ≥ 0。斜率可从 −A/B 得出。
To find the equation given two points, first compute m then use point-gradient form.
已知两点求直线方程，先计算斜率 m，再代入点斜式。
To convert between forms, rearrange algebraically.
形式之间可通过代数变形互相转换。
4. Parallel and Perpendicular Lines | 平行线与垂直线
Two lines are parallel if and only if their gradients are equal: m₁ = m₂. Their equations can be written in the forms y = mx + c₁ and y = mx + c₂.
两条直线平行当且仅当它们的斜率相等：m₁ = m₂。它们的方程可写成 y = mx + c₁ 和 y = mx + c₂ 的形式。
Two lines are perpendicular if the product of their gradients is −1: m₁ × m₂ = −1, provided neither line is vertical. If one line is vertical (undefined gradient), the perpendicular line is horizontal (gradient 0).
两条直线垂直当且仅当它们的斜率乘积为 −1：m₁ × m₂ = −1，前提是没有垂直线。若一条线为垂直线（无定义斜率），则垂直线为水平线（斜率为 0）。
When finding the equation of a line parallel or perpendicular to a given line through a given point, use the appropriate gradient and the point-gradient form.
求过给定点且与已知直线平行或垂直的直线方程时，使用相应的斜率并代入点斜式即可。
5. Perpendicular Bisectors | 垂直平分线
The perpendicular bisector of a segment AB passes through the midpoint of AB and is perpendicular to AB.
线段 AB 的垂直平分线经过 AB 的中点且垂直于 AB。
To find its equation: (1) calculate midpoint M, (2) find gradient m_AB, (3) gradient of perpendicular bisector = −1/m_AB, (4) use point-gradient form at M.
求其方程的步骤：(1) 计算中点 M，(2) 求出 AB 的斜率 m_AB，(3) 垂直平分线的斜率为 −1/m_AB，(4) 在点 M 处使用点斜式。
This concept is useful for locating the centre of a circle passing through three points: the perpendicular bisectors of two chords meet at the centre.
该概念可用于求经过三点的圆的圆心：两条弦的垂直平分线的交点即为圆心。
6. Distance from a Point to a Line | 点到直线的距离
The perpendicular distance d from a point (x₁, y₁) to the line Ax + By + C = 0 is given by: d = |Ax₁ + By₁ + C| / √(A² + B²).
点 (x₁, y₁) 到直线 Ax + By + C = 0 的垂直距离 d 为：d = |Ax₁ + By₁ + C| / √(A² + B²)。
Ensure the line equation is in general form with integer coefficients before applying the formula.
使用公式前请确保直线方程为一般式，且系数为整数。
This formula is widely used in circle geometry to determine whether a line is a tangent (distance = radius), secant (distance < radius), or does not intersect (distance > radius).
该公式在圆的几何中广泛应用，用于判断直线与圆的位置关系：相切（距离 = 半径）、相交（距离 < 半径）或不相交（距离 > 半径）。
7. Intersection of Two Lines | 两条直线的交点
To find the intersection point of two lines, solve their equations simultaneously. This can be done by substitution or elimination.
求两条直线的交点，需联立它们的方程求解。可通过代入法或消元法完成。
If the lines are given in different forms, convert them to a common form first. The solution (x, y) gives the coordinates of the crossing point.
若两直线方程形式不同，先化为同一种形式。解得的 (x, y) 即为交点坐标。
If the lines are parallel they will never intersect (no solution); if they coincide there are infinitely many solutions.
若两直线平行，则永不相交（无解）；若两直线重合，则有无限多解。
8. Area of a Triangle and Collinearity | 三角形面积与共线条件
The area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is |½ [ x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) ]|.
顶点为 (x₁, y₁), (x₂, y₂), (x₃, y₃) 的三角形面积为 |½ [ x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) ]|。
If the area equals zero, the three points are collinear (lie on a straight line). This provides a quick test for collinearity without finding gradients.
若面积为零，则三点共线（在一条直线上）。这提供了一个无需计算斜率即可快速检验共线的方法。
This formula is also useful when a quadrilateral is divided into two triangles to compute its area.
该公式也可用于将四边形分割为两个三角形从而计算面积。
9. Equation of a Circle | 圆的方程
A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r² (standard form). The general form is x² + y² + 2gx + 2fy + c = 0, where centre = (−
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