Ashley Jones
Assignment 9: Pedal Triangles
For this write-up I chose to take a look at what happens to the pedal triangle when the pedal point, P, is located at various centers of the original triangle ABC. Using GSP I was able to explore the pedal triangles and came with some mathematical conclusions.
First I looked at the pedal triangle when the pedal point, P, is located at the centroid. Before examining this pedal triangle I thought about the centroid. The centroid is the intersecting point of a triangle's three medians. This point, G, is always located inside the triangle. It does not matter if the triangle is acute, right, or obtuse, the centroid will always be found inside the triangle. Knowing this, I assumed that if the pedal point, P, is located at the centroid then the pedal triangle will be located inside the original triangle. Thus, the vertices of the pedal triangle will lie on the three sides of the original triangle. Using GSP I explored this idea. Below are some images from my exploration that verify my hypothesis.
In addition, if you would like to vary the triangle ABC for yourself to see how it affects the pedal triangle when the pedal point is located at the centroid, then I have included my GSP file for the exploration.
Next I looked at the pedal triangle when the pedal point, P, is located at the incenter. Before examining this pedal triangle I thought about the incenter of any triangle ABC. The incenter is the intersection point of the three angle bisectors of the triangle. Again, the incenter is always found in the interior of the original triangle. Therefore, I suspected that the pedal triangle will be found within the original triangle when the pedal point, P is located at the incenter. If this was to be true than the vertices of the pedal triangle should lie on the three sides of the original triangle. Using GSP I explored this idea. Below are some images from my exploration that verify my hypothesis.
If you would like to vary the triangle for yourself to explore the pedal triangle when the pedal point is located at the incenter, then I have included my GSP file for your exploration.
The last center that I examined was the orthocenter. I wanted to see where the pedal triangle would be located if the pedal point, P, was located at the orthocenter of the original triangle. As before, I thought about the orthocenter's location before jumping into the pedal triangle exploration. I knew that the orthocenter is the intersection of the three altitudes of a triangle. However, unlike the previous centers, the orthocenter is not always located in the interior of the triangle. When the triangle is obtuse the orthocenter is located outside of the triangle. I hypothesized then that the pedal triangle would also be located outside of the original triangle. When the triangle is acute the orthocenter is located inside the triangle. I then hypothesized that the pedal triangle would be located inside the original triangle in this case. Lastly, when the triangle is a right triangle, the orthocenter is located on a vertex of the triangle. For this situation, I was unsure what the pedal triangle would look like and where it would be located.
Here are some snapshots from GSP when the pedal point, P, was located at the orthocenter of an obtuse triangle. I did find that that the pedal triangle (red) was outside of the original triangle. But, I also found that no matter what obtuse triangle you used, one of the vertices of the pedal triangle was located on a side of the original triangle.
Here are some snapshots from GSP when the pedal point, P, was located at the orthocenter of an acute triangle. I did find that the pedal triangle (red) was inside the original triangle. Each one of the vertices of the pedal triangle was on a side of the original triangle.
Here is a snapshot from GSP when the pedal point, P, was located at the orthocenter of an acute triangle. The orthocenter, and point P were found at the vertex of the right angle. I was surprised to find that when this was the case the pedal triangle did not exist. there was simply a line segment from the point P to the hypotenuse of the right triangle.
If you would like to explore when the pedal point, P, is located at the orthocenter any further I have included my GSP file for your exploration.
Pedal Triangle: P at Orthocenter
I found this exploration to be very interesting and thought provoking. I have not worked with the centers of triangles for quite a while, and was interested to see how they were connected to the pedal triangle of various triangles. The location of pedal point, P, is an exploration that would be easy to do in the classroom as well. As someone who has not entered the classroom yet, I am always looking for activities and explorations such as this to add to my library for future reference.