Assignment 9: Pedal Triangles

Presented by: Amanda Oudi

For this write-up, I will take a look at what happens if the pedal point P is located at the different centers of triangle ABC. Using GSP, I can easily explore how the location of the pedal point P varies as the triangle centers vary, and draw a few mathematical conclusions.

Case i: What if the pedal point P is the centroid of the triangle ABC.

Let’s begin by making a few initial conjectures. We know that the centroid, G, of a triangle is the point in which the the medians of a triangle intersect. As the shape of the triangle varies (acute, right, and obtuse), the centroid always remains inside the triangle.This makes sense since the centroid is referred to as the balance point of the triangle. With this in mind, I will make the guess that if the pedal point P is located at the centroid of the triangle, then the point will also remain inside the triangle. So, the vertices of the pedal triangle will lie on the sides of the original triangle ABC. Using GSP, I explored this conjecture and found that the vertices of the pedal triangle were on the sides of triangle ABC and that the pedal point always remained on the inside of the triangle. See GSP sketch for exploration.

Case ii: What if the pedal point P is the incenter of the triangle ABC.

Again, let’s make a few intial conjectures. We know that the incenter, I, is the point in which the angle bisectors of a triangle intersect. The incenter is always found inside a triangle. So again, I predict that the pedal point, P, will also lie within the triangle, and that the vertices of the pedal triangle lie on the sides of the original triangle ABC. Using GSP, I explored this conjecture and found that this was indeed the case. See GSP Sketch for exploration.

Case iii: What if the pedal point P is the orthocenter of the triangle ABC.

What’s the prediction this time? We know that the orthocenter, H, is the point in which the lines containing the altitudes of a triangle intersect. Unlike the centroid and incenter, the location of the orthocenter is not restricted to the inside of a triangle. Instead, the location of the orthocenter varies as the shape of the triangle varies. When the triangle is acute, the orthocenter is located inside the triangle. So, I expect that the pedal triangle will be located inside the triangle, as well as pedal point P. When the triangle is right, the orthocenter is located on the vertex of the right angle. I predict that the pedal triangle and point would lie inside the original triangle ABC. Lastly, when the triangle is obtuse, the orthocenter is located outside the triangle. So, I expect the pedal triangle will be located outside the triangle, as well as the pedal point.

Using GSP, I explored this conjecture and noted the following results:

When triangle ABC was an acute triangle, the pedal point and triangle was inside triangle ABC. The pedal point P was located at the orthocenter of an acute triangle. When triangle ABC was a right triangle, the pedal triangle did not exist! That was surprising! Instead, a segment was formed and the pedal point P was located at the vertex of the right angle, same as the orthocenter of a right triangle. See GSP sketch for exploration.

I predict somewhat similar results when pedal point P is the circumcenter of triangle ABC. Try it yourself!

What a thought provoking exploration. The mysterious question is why when the pedal triangle in case iii. is simply a line segment. Conjectures? Proofs? I need to think about this some and see I can come up with anything. This would be a good exploration to enhance the unit relating to triangle centerds. While triangle centers is part of the mathematics curriculum, not enough emphasis or explorations are done with these centers. The focus is more on the construction of 4 basic triangle centers: centroid, incenter, orthocenter, and circumcenter, rather than the interesting characteristics and relationships among the triangle centers.