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This lesson involves a well-known center of a triangle called the orthocenter.

Its definition and properties will be discussed, and an example will be worked showing how to find its location on a graph.

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The Orthocenter

If you had to find the center of a triangle, how would you do it? Well, it would depend on which center you were trying to find. Yes, there is more than one center to a triangle. As a matter of fact, there are many, many centers, but there are four that are most commonly discussed: the circumcenter, the incenter, the centroid, and the orthocenter. The center we are going to discuss in this lesson is the orthocenter.

Definitions

Let’s begin with a basic definition of the orthocenter.

The orthocenter is the point of concurrency of the three altitudes of a triangle. Since a triangle has three vertices, it also has three altitudes. An altitude is defined as a perpendicular segment drawn from the vertex of a triangle to the line containing the opposite side.

A point of concurrency is the point of intersection of three or more lines.

Properties and Diagrams

There are three types of triangles with regard to the angles: acute, right, and obtuse. When we are discussing the orthocenter of a triangle, the type of triangle will have an effect on where the orthocenter will be located.

Take a look at the following diagrams. Typically, we would expect the center of a triangle to be inside of it. Indeed, the orthocenter will always be inside of an acute triangle.

Triangle 1

However, the orthocenter is on a right triangle (specifically at the vertex of the right angle). The only way to draw a segment from point B to point C is to travel along one of the sides of the triangle.

The same is true when going from point A to point C.

Triangle 2

Finally, the orthocenter will be outside of an obtuse triangle (specifically opposite the longest side). The only way to draw a segment from point B to the opposite side and it also be perpendicular is to extend side AC. The same is true when drawing the segment from point A to side BC. This forces the point of concurrency to be outside the triangle.

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