Investigating Orthocenters of Triangles

By: Kimberly Young

An orthocenter of a triangle is the common intersection of three lines containing the altitudes. To construct an Orthocenter of a triangle, using GSP, begin with a triangle. Then construct the altitude for each angle and side. In the picture below, H is Orthocenter.



We can look at how the Orthocenter changes as the size of the triangle varies. Click here. With the point labeled drag, explore how the orthocenter changes as the size of the triangle changes. Are there any conclusions you can draw?

There are several neat things that happen when we change the size of the triangle. Lets take a look at an obtuse, acute, and right triangle and compare their orthocenters. In all the pictures below, H is the orthocenter.

Acute Triangle


Obtuse Triangle



Right Triangle:


There are three distinctions in the orthocenter of each of the triangles above. The acute triangle's orthocenter is with in the triangle, while the obtuse triangle's is out side of the triangle and an extension of the altitude. The right triangle's orthocenter is on the vertex of the right angle.

By looking at the tracing of the orthocenter, we can see something interesting about the orthocenter of the triangle as the size of the triangle changes. Click HERE. After viewing this, you may see that the trace of the orthocenter forms a shape similar to a parabola as the size of the triangle changes.