**Assignment 8 :: Altitudes and Orthocenters**

**By Jamie K.York****
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The altitude, or height, of a triangle is a component and
measurement that is taught very early in mathematics, as it is an essential
part of calculating area, A = 1/2 bh.
Constructing the altitude from each side of a given triangle gives us what we
refer to as the **orthocenter**,
the intersection of the three altitudes. Below is an example using a given
triangle ABC.

To further our investigation of triangles and orthocenters, continue with the steps below:

1. Construct any triangle ABC.

2.
Construct the *Orthocenter* H of triangle ABC.

3.
Construct the *Orthocenter* of triangle HBC.

4.
Construct the *Orthocenter* of triangle HAB.

5.
Construct the *Orthocenter* of triangle HAC.

6.
Construct the *Circumcircles*
of triangles ABC, HBC, HAB, and HAC.

The resulting image should appear similar to the one below.

Notice that in locating the orthocenters of the three triangles formed with point H
aren't new points, but they are in fact vertices of the original triangle ABC.
As we consider the midpoints of the segments of each triangle, we begin to see
the formation of the nine point circle that we worked on in assignment
4. We can complete the nine point circle in this example in the same manner
as we did in assignment 4 (see below).

Use the GSP file to further investigate different types of triangles
and the resulting configuration. Specifically compare and contrast actue, right, and obtuse triangles.

In assignment 5 we created a script tool that can be used to find the orthocenter of any given triangle. Use this tool to furthe investigate the concepts of altitudes and orthocenters.

Orthocenter Script Tool (GSP file)

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