Bill Tankersley's EMT 668 Page
Write-up 4
Assignment 4 - Problem 2


Assignment: Use GSP to construct an Orthocenter (H) and explore its location for various shapes of triangles.


The Orthocenter (H) of a triangle is the common intersection of the three lines containing the altitudes. What we will examine in this write-up is the location of the orthocenter for various types of triangles, i.e., acute, right, and obtuse triangles.

First of all, let's investigate the location of the orthocenter in acute triangles. Below, we see two acute triangles both with orthocenter (H). After dragging and experimenting with the location of each of the vertices , I noticed that the orthocenter remained within the boundaries of the triangle as long as the triangle remained acute.





In the second triangle above, notice that the angle at the bottom left of the triangle is close to being a right angle. Also, notice that the orthocenter is fairly close to that angle. After experimenting with several triangles of the same shape, I began to wonder about the location of the orthocenter on right triangles.

Below, we see two right triangles and their orthocenters:







After observing many right triangles and the location of their orthocenters, I feel safe in claiming that the orthocenter of a right triangle will always coincide with the vertex of the right angle.

Now, we are interested in looking at some obtuse triangles and the locations of their orthocenters. Since the orthocenters in right triangles moved to a vertex, I began to wonder if the orthocenters in obtuse triangles would be located at some point outside the triangle.

Here are two obtuse triangles with their orthocenters:






After observing several examples of obtuse triangles, I feel confident in making the statement that the orthocenter of an obtuse triangle lies on the extension of the altitude to the side of the triangle that is opposite the obtuse angle.

Click here to download the GSP file for the images above. Drag and click on a vertex to change the shape of the triangle and notice the different positions of the Orthocenter.


So, to summarize our findings, We make the following statements:

1. For an acute triangle, the orthocenter (H) will always lie at some point on the interior of the triangle.
2. For a right triangle, the orthocenter will lie on the vertex of the right angle.
3. For an obtuse triangle, the orthocenter will lie on the extension of the altitude to the side opposite the obtuse angle.



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