📚 Coordinate Geometry: Lines, Circles and Solving Problems | 坐标几何：直线、圆与问题求解
Coordinate geometry, also called analytic geometry, describes geometric objects using algebraic equations on the Cartesian plane. In Cambridge IGCSE and O Level Mathematics, you are expected to calculate distances, midpoints and gradients, to find equations of straight lines and circles, and to interpret intersections and tangents. This revision guide brings together the essential methods and common exam traps.
坐标几何，又称解析几何，在笛卡尔平面上用代数方程来描述几何对象。在剑桥 IGCSE 和 O Level 数学中，要求你能够计算距离、中点和斜率，求直线与圆的方程，并理解交点和切线。本复习指南将核心方法与常见考试陷阱集中讲解。
1. The Cartesian Plane and Coordinates | 笛卡尔平面与坐标
The Cartesian plane is formed by a horizontal x-axis and a vertical y-axis crossing at the origin O(0,0). Every point is written as an ordered pair (x, y), where x is the horizontal displacement and y is the vertical displacement from the origin.
笛卡尔平面由水平的 x 轴和竖直的 y 轴组成，它们在原点 O(0,0) 相交。每个点都写成有序数对 (x, y)，其中 x 是从原点出发的水平位移，y 是竖直位移。
Coordinates can be positive, negative or zero, depending on the quadrant. In the first quadrant both x and y are positive; in the second quadrant x is negative and y is positive; in the third both are negative; in the fourth x is positive and y is negative.
坐标可以为正、为负或为零，取决于所在象限。第一象限中 x 和 y 均为正；第二象限中 x 为负、y 为正；第三象限中两者均为负；第四象限中 x 为正、y 为负。
2. Distance Between Two Points | 两点间距离
The distance between two points A(x₁, y₁) and B(x₂, y₂) is found by applying Pythagoras' theorem to the horizontal and vertical differences. The formula is:
AB = √((x₂ - x₁)² + (y₂ - y₁)²)
两点 A(x₁, y₁) 和 B(x₂, y₂) 之间的距离，可通过对水平差和竖直差应用毕达哥拉斯定理求得。公式为：AB = √((x₂ - x₁)² + (y₂ - y₁)²)
For example, the distance from (1, 2) to (4, 6) is √((4-1)² + (6-2)²) = √9 + 16 = √25 lways subtract in the same order for both coordinates because the square removes the sign.
例如，从点 (1, 2) 到点 (4, 6) 的距离为 √((4-1)² + (6-2)²) = √9 + 16 = √25 = 5 个单位。两个坐标相减要保持相同顺序，因为平方会消除符号。
3. Midpoint of a Line Segment | 线段中点
The midpoint M of the line segment joining A(x₁, y₁) and B(x₂, y₂) is the average of the x-coordinates and the average of the y-coordinates. The formula is:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
连接 A(x₁, y₁) 和 B(x₂, y₂) 的线段的中点 M，是 x 坐标的平均值和 y 坐标的平均值。公式为：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
he midpoint is useful for proving that a quadrilateral is a parallelogram, since the diagonals of a parallelogram bisect each other.
如果 A 是 (3, -2)，B 是 (7, 4)，则中点为 ((3+7)/2, (-2+4)/2) = (5, 1)。中点在证明四边形为平行四边形时很有用，因为平行四边形的对角线互相平分。
4. Gradient of a Straight Line | 直线的斜率
The gradient m measures the steepness of a line and is defined as the change in y divided by the change in x. For two points (x₁, y₁) and (x₂, y₂):
m = (y₂ - y₁) / (x₂ - x₁)
斜率 m 衡量直线的倾斜程度，定义为 y 的变化量除以 x 的变化量。对于两点 (x₁, y₁) 和 (x₂, y₂)：m = (y₂ - y₁) / (x₂ - x₁)
tical line. When x₂ = x₁, the denominator is zero, so the vertical line equation is x = k.
正斜率从左到右上升，负斜率下降，零斜率对应水平直线，未定义斜率对应竖直直线。当 x₂ = x₁ 时分母为零，因此竖直直线方程是 x = k。
5. Equations of Straight Lines | 直线方程
e gradient-intercept form is y = mx + c, where m is the gradient and c is the y-intercept,
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