What are altitudes and orthocenters? These are both terms that are used frequently when dealing with triangles. They are defined as follows:
How does one construct an orthocenter? Figure 1 illustrates how the orthocenter can be found.
Now that the orthocenter is defined, let us find the orthocenter of the three interior triangles, triangles HAB, HBC, and HAC. Looking at Figure 1, where would you expect the orthocenters of triangles HAB, HBC, and HAC to be located? Figure 2 identifies these three other orthocenters. In the figure, H-HAB corresponds to the orthocenter of triangle HAB, H-HBC corresponds to the orthocenter of triangle HBC and H-HAC corresponds to the orthocenter of triangle HAC. These three new orthocenters are at the vertices of the original triangle. Notice also that these new orthocenters are located at the vertice of the original triangle ABC that is not one of the vertices of the triangle that you are not finding the orthocenter of(i.e. the orthocenter of triangle HBC is located at vertice A of the original triangle ABC).
By using the definition of an orthocenter, one can understand why the orthocenters of the three triangles containing the orthocenter of the original triangle ABC as one of its vertices are located at the vertices of the original triangle.
Let's now construct the nine point circle for each of the four triangles that we have been talking about. A nine point circle is a circle with the following nine points on its circumference:
Figure 3 illustrates the nine point circle. This figure is not the nine point circle that corresponds to the triangle ABC from above.
Keeping in mind this last figure and the definitions of the nine points listed above, how would you expect the nine point circles pertaining to triangles ABC, HAB, HAC and HBC to be positioned? Once you have developed a hypothesis, go to the nine point circle page to see the solution.
The explorations here are only a few of those that deal with altitudes and orthocenters. If the reader would like to do some further investigations into altitudes and orthocenters click here to go to GSP.