📚 The Midpoint Formula | 中点公式
The midpoint of a line segment is the point exactly halfway between its two endpoints. It is one of the most straightforward yet essential tools in coordinate geometry. For A-Level Mathematics (AQA), you need to know the formula, understand its derivation, and apply it confidently in both two and three dimensions.
一条线段的中点，就是位于两个端点正中间的点。这是坐标几何中最简单却最重要的工具之一。在A-Level数学（AQA）中，你需要掌握中点公式，理解其推导过程，并能熟练地在二维和三维空间中应用。
1. Definition of Midpoint | 中点的定义
the segment AB is the point that divides AB into two equal parts. In other words, AM = MB.
在笛卡尔平面中，给定两点 A(x₁, y₁) 和 B(x₂, y₂)，线段 AB 的中点 M 就是将 AB 分成相等两部分的点。也就是说，AM = MB。
Geometrically, the midpoint has an elegant property: its coordinates are simply the averages of the corresponding coordinates of the endpoints.
从几何角度来说，中点有一个优美的性质：它的坐标就是两个端点对应坐标的平均值。
2. The Midpoint Formula (2D) | 中点公式（二维）
For points A(x₁, y₁) and B(x₂, y₂), the midpoint M has coordinates:
对于点 A(x₁, y₁) 和 B(x₂, y₂)，中点 M 的坐标为：
M = ( (x₁ + x₂)/2, (y₁ + y₂)/2 )
This formula works for any real values of x and y, including negative numbers and fractions. It is worth memorising because it appears frequently in coordinate geometry problems.
这个公式适用于任意实数 x 和 y，包括负数和分数。由于它在坐标几何问题中经常出现，值得熟练掌握并记忆。
3. Derivation: Average the Coordinates | 推导：坐标取平均
Why do we average the coordinates? Imagine moving from point A to point B. The horizontal displacement is x₂ − x₁, and the vertical displacement is y₂ − y₁. To reach the midpoint, you move half of each displacement from A.
为什么要取平均坐标？想象一下从点 A 移动到点 B。水平位移是 x₂ − x₁，垂直位移是 y₂ − y₁。要到达中点，你需要从 A 出发，分别走这两段位移的一半。
= ½x₁ + ½x₂ = (x₁ + x₂)/2. Similarly for the y-coordinate: y₁ + ½(y₂ − y₁) = (y₁ + y₂)/2.
从 A 出发：x₁ + ½(x₂ − x₁) = ½x₁ + ½x₂ = (x₁ + x₂)/2。y 坐标同理：y₁ + ½(y₂ − y₁) = (y₁ + y₂)/2。
Thus the midpoint is simply the arithmetic mean of each coordinate pair.
因此，中点实际上就是每一对坐标的算术平均值。
4. Worked Example 1: Finding the Midpoint | 例1：求中点
Find the midpoint M of the segment joining A(2, 5) and B(8, 7).
求连接 A(2, 5) 和 B(8, 7) 的线段的中点 M。
Apply the formula:
套用公式：
M = ( (2 + 8)/2, (5 + 7)/2 ) = (10/2, 12/2) = (5, 6)
So the midpoint is (5, 6). You can check that the distance from (2,5) to (5,6) equals the distance from (5,6) to (8,7).
因此中点为 (5, 6)。你可以验证，从 (2,5) 到 (5,6) 的距离等于从 (5,6) 到 (8,7) 的距离。
5. Worked Example 2: Locating an Endpoint | 例2：求另一端点
Sometimes you are given the midpoint and one endpoint, and you must find the other endpoint.
有时题目会给出中点和其中一个端点，要求你求另一个端点。
Suppose M(4, −1) is the midpoint of A(2, 5) and B(x, y). Find B.
已知 M(4, −1) 是 A(2, 5) 和 B(x, y) 的中点，求 B。
Write two equations:
列出两个方程：
(2 + x)/2 = 4 and (5 + y)/2 = −1
Solve: 2 + x = 8 → x = 6. Then 5 + y = −2 → y = −7. So B = (6, −7).
解得：2 + x = 8，所以 x = 6；5 + y = −2，所以 y = −7。因此 B = (6, −7)。
This reverse method is especially useful in vector and circle geometry problems.
这种逆向方法在向量和圆相关几何问题中尤其常用。
6. The Midpoint in 3D | 三维中的中点
The midpoint formula extends naturally to three dimensions. For A(x₁, y₁, z₁) and B(x₂, y₂, z₂), the midpoint is:
中点公式可以自然推广到三维空间。对于 A(x₁, y₁, z₁) 和 B(x₂, y₂, z₂)，中点为：
M = ( (x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2 )
There is no extra difficulty; you simply average the z-coordinates as well.
这并不增加额外难度，只需对 z 坐标同样取平均即可。
For example, the midpoint of (1, 2, 3) and (5, 8, 7):
例如，求 (1, 2, 3) 和 (5, 8, 7) 的中点：
M = ( (1+5)/2, (2+8)/2, (3+7)/2 ) = (3, 5, 5)
7. Distance and Midpoint Interplay | 距离与中点的关系
The midpoint and the distance formulas are closely related. If the midpoint splits a segment equally, then the distances from each endpoint to the midpoint are equal. Conversely, if a point lies on AB and is equidistant from A and B, it must be the midpoint.
中点公式与两点间距离公式密切相关。如果中点将一条线段平分，那么两端点到中点的距离必然相等。反过来，如果某个点在线段 AB 上，并且到 A、B 的距离相等，那么它一定是中点。
This interplay is often used in problems involving circles, where the midpoint of a diameter is the circle centre, and the distance from the centre to either endpoint is the radius.
这种关系常被用于圆的题目中：直径的中点就是圆心，圆心到任一端点的距离就是半径。
Example: A(2, −1) and B(6, 5) are endpoints of a diameter. Find the centre.
例：A(2, −1) 和 B(6, 5) 是直径的两个端点，求圆心。
Centre = ( (2+6)/2, (−1+5)/2 ) = (4, 2)
8. Applications of the Midpoint | 中点的应用
The midpoint formula is not just a classroom exercise; it has many practical and mathematical uses:
中点公式不仅仅是课堂练习，它还有许多实际和数学上的用途：
- Finding the centre of a circle from the diameter endpoints. | 由直径端点求圆心。
- Dividing a line segment into two equal parts in coordinate geometry. | 在坐标几何中把线段分成两等份。
- Computing the point of symmetry in transformations such as reflection. | 计算反射等变换中的对称点。
- Working with median lines of triangles: the midpoint of a side is needed to draw the median. | 处理三角形中线：绘制中线需要先求边的中点。
- In physics, finding the midpoint of a force vector or a center of mass between two point masses. | 在物理中，求两个质点之间的质心或力向量中点。
Understanding these applications helps you transfer the formula across different topics in the AQA specification.
理解这些应用可以帮助你在AQA考纲的不同主题之间灵活迁移公式。
9. Common Mistakes and Tips | 常见错误与提示
Students often mix up the midpoint formula with the distance formula. Remember: midpoint uses addition and division by 2; distance uses subtraction, squares, and a square root.
同学们经常把中点公式和距离公式搞混。记住：中点公式是相加后除以2；距离公式是相减、平方、再开根号。
Other common errors:
其他常见错误：
- Forgetting to divide both coordinates by 2. | 忘记把两个坐标都除以2。
- Sign errors when coordinates are negative. For example, (−3 + 5)/2 is correct, not −3 + 5/2. | 坐标为负数时出现符号错误。例如，(−3 + 5)/2 是正确的，而不是 −3 + 5/2。
- Confusing the midpoint with the point of intersection of diagonals. Both use the same averaging idea, but in different contexts. | 把中点与对角线交点混淆。两者虽然都用取平均的思想，但应用背景不同。
Tip: Always write the coordinate pairs as (x₁, y₁) and (x₂, y₂) before substituting. This reduces careless mistakes.
提示：代入公式前，先将端点明确标记为 (x₁, y₁) 和 (x₂, y₂)。这样可以减少粗心错误。
10. Practice Questions | 练习题
Test your understanding with these AQA-style questions:
用下面的AQA风格题目测试你的理解：
| Question | 题目 | Answer | 答案 |
|---|---|
| 1. Find the midpoint of (3, −4) and (9, 8). | (6, 2) |
| 2. The midpoint of (a, 3) and (5, b) is (2, 6). Find a and b. | a = −1, b = 9 |
| 3. A diameter AB has midpoint (1, 4). If A = (2, 0), find B. | B = (0, 8) |
| 4. Find the midpoint of (−2, 7, 1) and (4, 3, 5). | (1, 5, 3) |
Practice helps you perform the operation quickly and spot patterns in more complex problems.
多做练习能帮助你快速完成运算，并在更复杂的问题中识别出规律。
11. Quick Reference | 快速参考
In two dimensions:
在二维空间中：
Midpoint M = ( (x₁ + x₂)/2, (y₁ + y₂)/2 )
In three dimensions:
在三维空间中：
Midpoint M = ( (x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2 )
Always ask: are we working in 2D or 3D? Add the z-coordinate average if needed.
始终确认：我们在二维还是三维中？需要的话，记得对 z 坐标取平均。
Once you master the midpoint formula, harder topics such as vector geometry, circle equations, and coordinate transformations become much easier.
一旦掌握了中点公式，后面更难的课题——如向量几何、圆方程和坐标变换——都会变得更加轻松。
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