Assignment 8
Altitudes and Orthocenters
Nicolina Scarpelli
In this assignment we are going to investigate the altitudes and orthocenters of triangles. Every triangle has three bases (any of its sides) and three altitudes (heights). Every altitude is the perpendicular segment from a vertex to its opposite side (or the extension of the opposite side). The altitude is also referred to as the height of a triangle.
It is interesting to note that in any triangle, the three lines containing the altitudes are concurrent at a single point, which may or may not be inside the triangle. Observe the three pictures below that I created using Geometer's Sketchpad.
As you can see, the picture above is a right triangle. Notice, the three altitudes meet at the vertex of the right triangle.
Here we have an acute triangle, and notice the altitudes are concurrent at a single point located inside the triangle.
Finally, we have an obtuse triangle and observe the three altitudes are concurrent at a single point located outside of the triangle.
As stated above, every triangle has three altitudes. The point at which all three altitudes meet is called the orthocenter.
Click here for the GSP file of the construction of the orthocenter.
Let's label the large triangle ABC and the orthocenter, H.
As you can see from the image above, the orthocenter splits the large triangle up into three individual triangles, BCH (the pink triangle below), ABH(the blue triangle below), and ACH(the green triangle below).
Now, let's find the orthocenters of each of these three triangles individually. This is shown below.
Now let's put them all together and and we obtain the sketch below.
Observe the picture above. What do you see? Notice the orthocenters of the three new triangles lie exactly on the vertices of the original triangle ABC. Observe that the orthocenter of ABH is at point C, the orthocenter of BCH is at point A, and the orthocenter of ACH is at point B. If you change the shape of the triangle, will the new orthocenters still remain on the vertices A, B, and C of the larger triangle ABC? Click here to explore different triangle shapes using the GSP construction.
Next, let's explore the circumcircles of both the original triangle ABC and the three smaller triangles, ABH, BCH, and ACH. I used my circumcircle script tool that I created in Assignment 5 to create the circumcircles of each triangle. Observe the circumcircles constructed below.
What do you see regarding the relationship between the circumcircles of each of the triangles? All of the circumcircles appear to be congruent. However, what will happen if any vertex of the triangle ABC was to move to where the orthocenter H is located? Where would H then be located? Click here or a GSP file.
If we move the vertex A of the triangle ABC to where the orthocenter H is located, then the purple circumcircle of the triangle ABC and the green circumcircle of the triangle BCH will overlap. Then H will be located at the position of vertex A. Since we move the vertex A of the triangle ABC to the vertex H of the triangle BCH, the purple circumcircle of the triangle ABC will move to the green circumcircle of the triangle BCH. Observe the image below that shows the overlapping of the circumcircles.
Let's now observe the changes when we move the vertex B of the triangle ABC to where the orthocenter H is located. When we do this the purple circumcircle of the triangle ABC and the red circumcircle of the triangle ACH will overlap. Then H will be located at the position of vertex B. Since we move the vertex B of the triangle ABC to the vertex H of the triangle ACH, the the purple circumcircle of the triangle ABC will move to the red circumcircle of the triangle ACH. Observe the image below that shows the overlapping of the circumcircles.
Finally, let's observe what happens when we move the vertex C of the triangle ABC to where the orthocenter H is located. When we do this, the purple circumcircle of the triangle ABC and the black circumcircle of the triangle ABH will overlap. Then H will be located at the position of vertex C. Since we move the vertex C of the triangle ABC to the vertex H of the triangle ABH, the purple circumcircle of the triangle ABC will move to the black circumcircle of the triangle ABH. Observe the image below that shows the overlapping of the circumcircles.
Next, let's construct the radii of each circumcircle. Observe the picture below to see that the four radii of the circumcircles are equal to each other. In addition, the segments AC1, BC1, BC2, CC2, CC3, and AC3 all have the same length. This is verified calculationally in the picture below.
Next, let's construct segments between C1C2, C2C3, and C3C1. We now have a triangle(shown in orange) that connects the circumcenters C1, C2, and C3. Observe the picture below.
Observe that the segment BC1 and the segment CC3 are parallel; thus the segment C1C3 is parallel to the segment BC. Since they are parallel, we can conclude that segment C1C3 has the same length as the segment BC. Similarly, segment AB is parallel to segment C2C3, so the length of segment AB is equal to the length of segment C2C3. Also, segment AC is parallel to segment C1C2; thus, the length of the segment AC equals the length of segment C1C2. Therefore, from this information we can conclude that triangle C1C2C3 is similar and congruent to triangle ABC. Click here for the interactive GSP construction.
A little fun fact! You might have already been able to observe this above, but when you connect the centers or the circumcircles (the circumcenters) with the nearest vertices of the triangles ABH, BCH, and ACH a cube is formed. Observe the picture below(the cube is shown in blue). Keep in mind that the shape of the cube will change whenever the vertices A, B, and C of the original triangle ABC are moved. Click here for the interactive GSP construction. Move the vertices and observe the changes.
This assignment was extremely helpful in developing Geometer's Sketchpad skills. These investigations would be useful in a high school geometry classroom in order to engage the students and help them learn the concepts of altitudes and orthocenters. While I observed in tenth grade geometry class in Knoxville, Tennessee the students seemed to have a tough time remembering the difference between the altitudes, the orthocenter, the centroid, the circumcenter, and the incenter. Many of the assignments on my web page investigate certain geometrical concepts that can help your students visualize what is happening. There are several more investigations that focus on the altitudes and orthocenters of triangles located on Dr. Jim Wilson's EMAT 6680 Webpage.
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