Orthocentre
Orthocentre
The orthocentre is a significant concept in the study of triangles in Euclidean geometry. It is one of the four main points associated with a triangle, the others being the centroid, circumcentre, and incentre.
Definition
The orthocentre of a triangle is the point where the three altitudes of the triangle intersect. An altitude of a triangle is a perpendicular line segment from a vertex to the line containing the opposite side (also known as the base).
Properties
- The orthocentre lies inside the triangle for an acute triangle, on the triangle for a right triangle (at the vertex of the right angle), and outside the triangle for an obtuse triangle.
- The orthocentre, centroid, circumcentre, and incentre of a triangle are collinear, lying on a line known as the Euler line.
- The orthocentre does not necessarily lie on the triangle's sides or within the triangle.
Formulas
The coordinates of the orthocentre can be found using the vertices of the triangle. If the vertices of the triangle are $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$, then the slopes of the altitudes can be found using the negative reciprocal of the slopes of the sides of the triangle.
For example, the slope of the line BC is given by:
$$ m_{BC} = \frac{y_3 - y_2}{x_3 - x_2} $$
The slope of the altitude from A is then:
$$ m_{A} = -\frac{1}{m_{BC}} = -\frac{x_3 - x_2}{y_3 - y_2} $$
Using the point-slope form of a line, the equation of the altitude from A can be written as:
$$ y - y_1 = m_{A}(x - x_1) $$
By finding the equations of the altitudes from two vertices and solving them simultaneously, the coordinates of the orthocentre can be determined.
Differences and Important Points
| Property | Orthocentre | Centroid | Circumcentre | Incentre |
|---|---|---|---|---|
| Definition | Intersection of altitudes | Intersection of medians | Intersection of perpendicular bisectors | Intersection of angle bisectors |
| Location for Acute Triangle | Inside the triangle | Inside the triangle | Outside the triangle | Inside the triangle |
| Location for Right Triangle | At the right angle vertex | Inside the triangle | At the midpoint of the hypotenuse | Inside the triangle |
| Location for Obtuse Triangle | Outside the triangle | Inside the triangle | Outside the triangle | Inside the triangle |
| Collinear with | Centroid, circumcentre, incentre (Euler line) | Orthocentre, circumcentre, incentre (Euler line) | Orthocentre, centroid, incentre (Euler line) | Orthocentre, centroid, circumcentre (Euler line) |
Examples
Example 1: Finding the Orthocentre of a Triangle
Given a triangle with vertices $A(2, 3)$, $B(4, 7)$, and $C(6, 3)$, find the orthocentre.
Solution:
First, find the slopes of the sides:
$$ m_{AB} = \frac{7 - 3}{4 - 2} = 2 $$ $$ m_{BC} = \frac{3 - 7}{6 - 4} = -2 $$ $$ m_{CA} = \frac{3 - 3}{6 - 2} = 0 $$
Now, find the slopes of the altitudes:
$$ m_{A} = -\frac{1}{m_{BC}} = \frac{1}{2} $$ $$ m_{B} = -\frac{1}{m_{CA}} = \text{undefined (vertical line)} $$ $$ m_{C} = -\frac{1}{m_{AB}} = -\frac{1}{2} $$
Since the altitude from $B$ is vertical, it has the equation $x = 4$. The equation of the altitude from $A$ using the point-slope form is:
$$ y - 3 = \frac{1}{2}(x - 2) $$
Solving the two equations $x = 4$ and $y - 3 = \frac{1}{2}(x - 2)$ simultaneously gives us the orthocentre $H(4, 4)$.
Example 2: Orthocentre of a Right Triangle
Given a right triangle with vertices $A(0, 0)$, $B(0, 4)$, and $C(6, 0)$, find the orthocentre.
Solution:
Since triangle ABC is a right triangle with the right angle at A, the orthocentre is at the vertex of the right angle, which is $A(0, 0)$.
These examples illustrate how to find the orthocentre of a triangle using the vertices and the properties of the altitudes. Understanding the orthocentre is essential for solving various geometric problems and proving theorems related to triangles.