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Orthocentre of a Triangle Properties
Orthocenter of a triangle formula: A triangle has three vertices, or corners, and three sides. The orthocentre is a special point in a triangle that is located at the intersection of the three altitudes. The altitudes are the lines that connect the vertices of a triangle to the opposite side. The orthocentre is also the centre of the triangle’s circumcircle.
Orthocenter Definition
The orthocenter of a triangle is the intersection of its three altitudes. It is also the center of the triangle’s nine-point circle.
Orthocenter Properties
There are three orthocenter properties:
- The orthocenter is the intersection of the three altitudes of a triangle.
- orthocenter is the center of the triangle’s nine-point circle.
- The orthocenter is also the center of the triangle’s Euler line.
How to Calculate Orthocenter of a Triangle :
The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. To calculate the orthocenter of a triangle, you need to know the lengths of the three altitudes.
Once you have the lengths of the altitudes, simply use the Pythagorean theorem to calculate the distance from each vertex to the orthocenter. The orthocenter will be the point where the three distances intersect.
There are a few different ways to calculate the orthocenter of a triangle. The most common method is to use the coordinates of the vertices of the triangle.
To find the orthocenter using the coordinates of the vertices, first find the equation of the line that contains the two vertices. This can be done by finding the slope of the line and then using the point-slope form of the equation of a line. Once the equation of the line is found, plug in the coordinates of the third vertex and solve for the y-coordinate. This will give you the y-coordinate of the orthocenter. To find the x-coordinate of the orthocenter, plug the y-coordinate into the equation of the line.
Another method to find the orthocenter is to use the altitudes of the triangle. The altitude of a triangle is the perpendicular line from a vertex to the opposite side. To find the orthocenter using the altitudes, find the equations of the altitudes. These can be found by finding the slope of the altitude and then using the point-slope form of the equation of a line. Plug in the coordinates of the other two vertices and solve for the y-coordinate. This will give you the y-coordinate of the orthocenter. To find the x-coordinate of the orthocenter, plug the y-coordinate into the equation of the line.
1. Find the orthocenter of the triangle with the given vertices:
A(-1, 2), B(3, 4), and C(5, -6)
The orthocenter is at the intersection of the three altitudes of the triangle. The altitudes are the lines from a vertex to the opposite side. The three altitudes are:
AB: From A to BC
AC: From A to CB
BC: From B to CA
