Pedal Triangles

by

Becky Mohl

 

Example 1: Pedal Point, P is located inside the triangle ABC. Explore using the Animation.

Example 2: Pedal point, P is located outside the triangle ABC. Explore using the Animation.

 

Now lets consider the cases when the Pedal triangle has one of the four special centers.

Centroid: Consider what we know about the centroid of any given triangle. We now that it is the intersection of the three medians. We also know that the centroid cannot ever be located outside of the triangle; therefore, if the Pedal Triangle has the centroid as its center it also cannot be located outside of the given triangle. Here is an Animation to explore farther. We see below that the triangle is acute, and the centroid is inside the triangle ABD; therefore, the Pedal triangle also lies inside the triangle. After exploring, I found that the Pedal triangle is always inside the triangle ABC if the centroid is the center.

 

 

Orthocenter: We know that the orthocenter is the intersection of the three altitudes of any given triangle. We also know that the orthocenter can be located outside of the triangle if the triangle is obtuse. However, can the Pedal Triangle lie outside the triangle if the center is the orthocenter Yes, it does. Use the Animation to explore. We see below if the triangle is acute, the orthocenter and Pedal Triangle both lie in the triangle ABC. After exploring, we see that when the triangle is obtuse, the orthocenter lies outside the triangle, and the majority of the Pedal Triangle is also outside the triangle. At some places the Pedal Triangle overlapped the triangle ABC, but it did always lie partially outside of triangle ABC.

 

Circumcenter: Again, we know that the circumcenter can lie outside of a given triangle; therefore, this means that the Pedal Triangle with the center the circumcenter can also lie outside the triangle. Here is an Animation to explore the possibility. The same holds true for the circumcenter. IF the triangle is acute as we see below, the circumcenter and the Pedal Triangle are both inside triangle ABC. After exploring with GSP, I found the same as the orthocenter. When the triangle ABC is obtuse, the Pedal triangle also lies wither outside of the triangle or partially inside and outside; however, it cannot be fully inside the triangle unless the triangle is acute.

 

Incenter: Like the centroid, we know that the incenter always lie inside the triangle; therefore, the Pedal Triangle with the incenter as the center will also lie inside of the triangle. Use the GSP Animation to see that the Pedal triangle always lies inside triangle ABC if the incenter is the center.

 

 

After considering these four cases, lets see what happens when the Pedal Triangle has a center aligned with the triangle ABC's nine point circle.

 

Now consider the case where the Pedal Point, P lies on one of the vertices of the triangle ABC.

 

....and on one of the sides of the triangle ABC.

 

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