正交中心公式
Orthocenter 属于几何的概念。正交一词意味着正确，它被认为是从三角形顶点绘制的所有三个高度的交点。正交中心是三角形所有高度的交点。
正交中心公式
垂心是三角形中海拔高度相交的点。高度是从三角形的一侧到相对顶点的垂直线。在下图中，Orthocenter 用字母“O”表示，高度为 AE、BF、CD，顶点为 A、B、C。
There is no particular formula to calculate the orthocenter of a triangle. But it can figure out by the below steps:
- Find the Slope of the sides of the triangle.
Slope of a side = (y2 – y1)/(x2 – x1)
Where (x1, y1) and (x2, y2) are endpoints of the side.
- To find the altitude slope which is perpendicular to the sides of the triangle can be calculated by,
Perpendicular Slope of a line (Altitude) = -1/(Slope of a side)
- To calculate the equation for altitude, the point-slope formula is,
(y – y1) = m × (x – x1)
So by finding any two slopes and equation of altitudes, the orthocenter can be calculated by solving those equations.
示例问题
问题 1：求两点 A(1, 10), B(5, 5) 之间直线的斜率。
解决方案：
Given A(1, 10), B(5, 5)
Slope Of line AB = (y2 – y1)/(x2 – x1)
= (5 – 10)/(5 – 1)
= -5/4
Slope of line AB = -5/4
问题 2：求两点 A(5, 4), B(-1, 2) 之间直线的斜率。
解决方案：
Given A(5, 4), B(-1, 2)
Slope Of line AB = (y2 – y1)/(x2 – x1)
= (2 – 4)/(-1 – 5)
= -2/-6
= 1/3
Slope of line AB = 1/3
问题3：如果AB的斜率为-5/4，求AB的垂直斜率？
解决方案：
Given Slope of AB = -5/4
Perpendicular Slope of AB = -1/Slope of AB
= -1/(-5/4)
= 4/5
Perpendicular slope of AB is 4/5.
问题4：求顶点为A(1, 0), B(3, 1), C(2, 3)时三角形的垂心。
解决方案：
Given Vertices:
A(1, 0), B(3, 1), C(2, 3)
Slope of AB = (y2 – y1)/(x2 – x1)
= (1 – 0)/(3 – 1)
= 1/2
Slope of CD = Perpendicular slope of AB
= -1/Slope of AB
= -1/(1/2)
= -2
The equation of cd is
(y – y1) = m × (x – x1)
(y – 3) = -2 × (x – 2)
y – 3 = -2x + 4
2x + y – 3 – 4 = 0
2x + y – 7 = 0 ⇢ (1)
Slope of AC = (y2 – y1)/(x2 – x1)
= (3 – 0)/(2 – 1)
= 3
Slope of BF = Perpendicular slope of AC
= -1/Slope of AC
= -1/3
The equation of BF is,
(y – y1) = m × (x – x1)
(y – 1) = (-1/3) × (x – 3)
3(y – 1) = -1(x – 3)
3y – 3 = -x + 3
x + 3y – 6 = 0 ⇢ (2)
Solve Equation (1) & (2)
(1) × 1 => 2x + y – 7 = 0
(2) × 2 => 2x + 6y – 12 = 0
-5y + 5 = 0
-5y = -5
y = 1
Substitute y value in equation – (1)
2x + y – 7 = 0
2x + 1 – 7 = 0
2x = 6
x = 3
Orthocenter is O(3, 1)
问题5：求顶点为A(0, 0), B(2, -1), C(1, 3)时三角形的垂心。
解决方案：
Given Vertices:
A(0, 0), B(2, -1), C(1, 3)
Slope of AB = (y2 – y1)/(x2 – x1)
= (-1 – 0)/(2 – 0)
= -1/2
Slope of CD = Perpendicular slope of AB
= -1/Slope of AB
= -1/(-1/2)
= 2
The equation of CD is,
(y – y1) = m × (x – x1)
(y – 3) = 2 × (x – 1)
y – 3 = 2x – 2
2x – y + 1 = 0 ⇢ (1)
Slope of AC = (y2 – y1)/(x2 – x1)
= (3 – 0)/(1 – 0)
= 3
Slope of BF = Perpendicular slope of AC
= -1/Slope of AC
= -1/3
The equation of BF is,
(y – y1) = m × (x – x1)
(y – (-1)) = (-1/3) × (x – 2)
3(y + 1) = -1(x – 2)
3y + 3 = -x + 2
x + 3y + 1 = 0 ⇢ (2)
Solve Equation (1) & (2)
(1) × 1 => 2x – y + 1 = 0
(2) × 2 => 2x + 6y + 2 = 0
-7y – 1 = 0
-7y = 1
y = -1/7
Substitute y value in equation-(1)
2x – y + 1 = 0
2x – (-1/7) + 1 = 0
2x + (1/7) + 1 = 0
2x = -1 – (1/7)
2x = (-7 – 1)/7
x = -8/14
x = -4/7
The orthocenter for the given data is O(-4/7, -1/7)
问题六：当顶点为A(3, 2), B(0, 3), C(-2, 1)时求三角形的垂心
解决方案：
Given Vertices:
A(3, 2), B(0, 3), C(-2, 1)
Slope of AB = (y2 – y1)/(x2 – x1)
= (3 – 2)/(0 – 3)
= -1/3
Slope of CD = Perpendicular slope of AB
= -1/Slope of AB
= -1/(-1/3)
= 3
The equation of CD is,
(y – y1) = m × (x – x1)
(y – 1) = 3 × (x + 2)
y – 1 = 3x + 6
3x – y + 7 = 0 ⇢ (1)
Slope of AC = (y2 – y1)/(x2 – x1)
= (1 – 2)/(-2 – 3)
= -1/-5
= 1/5
Slope of BF = Perpendicular slope of AC
= -1/Slope of AC
= -1/(1/5)
= -5
The equation of BF is
(y – y1) = m × (x – x1)
(y – 3) = (-5) × (x – 0)
y – 3 = -5x
5x + y – 3 = 0 ⇢ (2)
Solve Equation (1) & (2)
(1) × 1 => 3x – y + 7 = 0
(2) × 1 => 5x + y – 3 = 0
8x + 4 = 0
8x = -4
x = -4/8
x = -1/2
Substitute x value in equation-(1)
3x – y + 7 = 0
3(-1/2) – y + 7 = 0
(-3/2) – y + 7 = 0
y = 7 – (3/2)
y = (14 – 3)/2
y = 11/2
The orthocenter for the given data is O(-1/2, 11/2)
问题7：如果斜率和数据点分别为-1/2，则求直线方程，(3, 3)。
解决方案：
Given (x1, y1) = (3, 3)
Slope (m) = -1/2
Equation of Line is,
(y – y1) = m × (x – x1)
(y – 3) = (-1/2) × (x – 3)
2(y – 3) = -1(x – 3)
2y – 6 = -x + 3
x + 2y – 9 = 0
The equation of line for the given slope and data point is x + 2y – 9 = 0.