What are altitudes and orthocenters? These are both terms that are used frequently when dealing with triangles. They are defined as follows:

**Altitude**--A perpendicular segment or line from a vertex to the line of the opposite side.**Orthocenter**--The common intersection of the three lines containing the altitudes. This normally labeled**H**.

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:

- The midpoint of segment AB
- The midpoint of segment AC
- The midpoint of segment BC
- The midpoint of segment HA
- The midpoint of segment HB
- The midpoint of segment HC
- The point of intersection between line AB and the altitude passing through point C
- The point of intersection between line AC and the altitude passing through point B
- The point of intersection between line BC and the altitude passing through point A

**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**.