教材内容 / Textbook Content
在坐标几何中，线段的中点是连接两个端点的线段的中点。我们可以通过平均两个端点的x坐标和y坐标来找到中点的坐标。
In coordinate geometry, the midpoint of a line segment is the point that divides the segment into two equal parts. We can find the midpoint coordinates by averaging the x-coordinates and y-coordinates of the endpoints.
中点 / Midpoint：具有端点 \(\left( x_1, y_1 \right)\) 和 \(\left( x_2, y_2 \right)\) 的线段的中点是 \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)。
The midpoint of a line segment with endpoints \(\left( x_1, y_1 \right)\) and \(\left( x_2, y_2 \right)\) is \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)。
中点公式是坐标几何中的基本工具，它告诉我们：要找到两点之间的中点，只需将相应的坐标相加然后除以2。
The midpoint formula is a fundamental tool in coordinate geometry. It tells us that to find the midpoint between two points, we simply add the corresponding coordinates and divide by 2.
中点公式 / Midpoint Formula：
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
其中 \(M\) 是中点，\((x_1, y_1)\) 和 \((x_2, y_2)\) 是线段的两个端点。
问题 / Problem：线段 \({AB}\) 是一个圆的直径，其中 \(A\) 和 \(B\) 分别是 \((-3, 8)\) 和 \((5, 4)\)。求圆心的坐标。
The line segment \({AB}\) is a diameter of a circle, where \(A\) and \(B\) are \((-3, 8)\) and \((5, 4)\) respectively. Find the coordinates of the centre of the circle.
解答 / Solution：
记住圆心是直径的中点。/ Remember the centre of a circle is the midpoint of a diameter.
圆心是 \(\left( \frac{-3 + 5}{2}, \frac{8 + 4}{2} \right) = \left( \frac{2}{2}, \frac{12}{2} \right) = (1, 6)\)
The centre of the circle is \(\left( \frac{-3 + 5}{2}, \frac{8 + 4}{2} \right) = \left( \frac{2}{2}, \frac{12}{2} \right) = (1, 6)\)
问题 / Problem：线段 \({PQ}\) 是一个圆心为 \((2, -2)\) 的圆的直径。已知 \(P\) 是 \((8, -5)\)，求 \(Q\) 的坐标。
The line segment \({PQ}\) is a diameter of the circle centre \((2, -2)\). Given that \(P\) is \((8, -5)\), find the coordinates of \(Q\).
解答 / Solution：
设 \(Q\) 的坐标为 \((a, b)\)。因为 \((2, -2)\) 是 \((a, b)\) 和 \((8, -5)\) 的中点：
Let \(Q\) have coordinates \((a, b)\). Since \((2, -2)\) is the midpoint of \((a, b)\) and \((8, -5)\):
\(\left( \frac{8 + a}{2}, \frac{-5 + b}{2} \right) = (2, -2)\)
分别比较x坐标和y坐标：/ Compare the x- and y-coordinates separately:
\(\frac{8 + a}{2} = 2 \Rightarrow 8 + a = 4 \Rightarrow a = -4\)
\(\frac{-5 + b}{2} = -2 \Rightarrow -5 + b = -4 \Rightarrow b = 1\)
因此，\(Q\) 是 \((-4, 1)\)。/ So, \(Q\) is \((-4, 1)\).
线段的垂直平分线是垂直于该线段并通过其中点的直线。垂直平分线在几何学中具有重要性质和应用。
The perpendicular bisector of a line segment is the straight line that is perpendicular to the segment and passes through the midpoint of the segment. Perpendicular bisectors have important properties and applications in geometry.
垂直平分线 / Perpendicular Bisector：线段 \({AB}\) 的垂直平分线是垂直于 \({AB}\) 并通过 \({AB}\) 中点的直线。
The perpendicular bisector of a line segment \({AB}\) is the straight line that is perpendicular to \({AB}\) and passes through the midpoint of \({AB}\).
如果线段 \({AB}\) 的斜率是 \(m\)，那么它的垂直平分线 \(l\) 的斜率将是 \(-\frac{1}{m}\)。
If the gradient of \({AB}\) is \(m\) then the gradient of its perpendicular bisector, \(l\), will be \(-\frac{1}{m}\).
垂直条件 / Perpendicular Condition：
如果两条直线互相垂直，它们的斜率满足：\(m_1 \cdot m_2 = -1\)
If two lines are perpendicular, their gradients satisfy: \(m_1 \cdot m_2 = -1\)
问题 / Problem：线段 \({AB}\) 是一个圆心为 \(C\) 的圆的直径，其中 \(A\) 和 \(B\) 分别是 \((-1, 4)\) 和 \((5, 2)\)。直线 \(l\) 通过 \(C\) 并垂直于 \({AB}\)。求 \(l\) 的方程。
The line segment \({AB}\) is a diameter of the circle centre \(C\), where \(A\) and \(B\) are \((-1, 4)\) and \((5, 2)\) respectively. The line \(l\) passes through \(C\) and is perpendicular to \({AB}\). Find the equation of \(l\).
解答 / Solution：
步骤1：求圆心（中点）/ Step 1: Find the centre (midpoint)
圆心 \(C = \left( \frac{-1 + 5}{2}, \frac{4 + 2}{2} \right) = (2, 3)\)
The centre of the circle is \(C = \left( \frac{-1 + 5}{2}, \frac{4 + 2}{2} \right) = (2, 3)\)
步骤2：求线段 \({AB}\) 的斜率 / Step 2: Find the gradient of \({AB}\)
\(m_{AB} = \frac{2 - 4}{5 - (-1)} = \frac{-2}{6} = -\frac{1}{3}\)
The gradient of the line segment \({AB}\) is \(m_{AB} = \frac{2 - 4}{5 - (-1)} = \frac{-2}{6} = -\frac{1}{3}\)
步骤3：求垂直平分线的斜率 / Step 3: Find the gradient of the perpendicular bisector
因为两条直线垂直，所以：\(m_l = 3\)（因为 \(-\frac{1}{3} \times 3 = -1\)）
Since the lines are perpendicular: \(m_l = 3\) (because \(-\frac{1}{3} \times 3 = -1\))
步骤4：求直线方程 / Step 4: Find the equation of the line
直线通过点 \((2, 3)\) 且斜率为 \(3\)，使用点斜式：
The line passes through \((2, 3)\) with gradient \(3\), using point-slope form:
\(y - 3 = 3(x - 2)\)
\(y - 3 = 3x - 6\)
\(y = 3x - 3\)
在求解垂直平分线方程时，必须确保：
1. 直线通过线段的中点
2. 直线的斜率与原线段的斜率满足垂直条件 \((m_1 \cdot m_2 = -1)\)
When finding the equation of a perpendicular bisector, ensure that:
1. The line passes through the midpoint of the segment
2. The gradient satisfies the perpendicular condition \((m_1 \cdot m_2 = -1)\)
为什么垂直平分线上的任意一点到线段两端点的距离相等？这个性质在实际问题中有哪些应用？
Why is any point on the perpendicular bisector equidistant from both endpoints of the segment? What applications does this property have in practical problems?
中点和垂直平分线的概念在解决复杂几何问题时经常需要结合使用。让我们通过一个综合例子来展示这些概念的应用。
The concepts of midpoints and perpendicular bisectors often need to be used together when solving complex geometric problems. Let's demonstrate the application of these concepts through a comprehensive example.
问题 / Problem：三角形 \({ABC}\) 的顶点分别为 \(A(1, 2)\)、\(B(7, 4)\) 和 \(C(3, 8)\)。求：
a) 边 \({AB}\) 的中点坐标
b) 边 \({AB}\) 的垂直平分线方程
c) 验证垂直平分线是否通过三角形的外心
Triangle \({ABC}\) has vertices at \(A(1, 2)\), \(B(7, 4)\), and \(C(3, 8)\). Find:
a) The midpoint of side \({AB}\)
b) The equation of the perpendicular bisector of \({AB}\)
c) Verify if the perpendicular bisector passes through the circumcenter
解答 / Solution：
a) 中点计算 / Midpoint calculation：
中点 \(M = \left( \frac{1 + 7}{2}, \frac{2 + 4}{2} \right) = (4, 3)\)
Midpoint \(M = \left( \frac{1 + 7}{2}, \frac{2 + 4}{2} \right) = (4, 3)\)
b) 垂直平分线方程 / Perpendicular bisector equation：
边 \({AB}\) 的斜率：\(m_{AB} = \frac{4 - 2}{7 - 1} = \frac{2}{6} = \frac{1}{3}\)
Gradient of \({AB}\): \(m_{AB} = \frac{4 - 2}{7 - 1} = \frac{2}{6} = \frac{1}{3}\)
垂直平分线的斜率：\(m_{\perp} = -3\)（因为 \(\frac{1}{3} \times (-3) = -1\)）
Gradient of perpendicular bisector: \(m_{\perp} = -3\) (since \(\frac{1}{3} \times (-3) = -1\))
通过点 \((4, 3)\) 且斜率为 \(-3\) 的直线方程：
Equation of line through \((4, 3)\) with gradient \(-3\):
\(y - 3 = -3(x - 4)\)
\(y - 3 = -3x + 12\)
\(y = -3x + 15\)
垂直平分线在三角形几何中具有特殊意义：三角形三条边的垂直平分线交于一点，这个点称为三角形的外心（circumcenter），也就是三角形外接圆的圆心。
Perpendicular bisectors have special significance in triangle geometry: the three perpendicular bisectors of a triangle's sides meet at a single point called the circumcenter, which is the center of the triangle's circumscribed circle.
在计算垂直平分线时，学生常犯的错误包括：
1. 忘记使用中点：垂直平分线必须通过线段的中点
2. 斜率计算错误：垂直斜率应该是原斜率的负倒数
3. 符号错误：在坐标运算中要特别注意正负号
通过本节的学习，你应该能够：
本节内容为后续学习奠定了重要基础：