Amberly Roberts

Assignment 9: Pedal Triangles

Any arbitrary point in the plane contained by a triangle can be the pedal point of a triangle. When constructing the lines perpendicular from the pedal point to the lines determined by the sides of the triangle, we get three points of intersection. By connecting these points of intersection, we get a triangle called the pedal triangle.

In the construction above, P is the pedal point of triangle ABC. Triangle DEF is the pedal triangle of triangle ABC.

In this investigation, we will explore the relationship between the location of the pedal point and the pedal triangle. Click here to open up a GSP file that will allow you to see how the pedal triangle DEF changes as you change the location of the pedal point P.

When investigating, I hope that you noticed the following things...

1. The pedal triangle can lie completely outside of the triangle, completely inside the triangle, or with part inside and part outside.

2. For some locations of the pedal point, the triangle dissapears!

Now, we will get specific. First we will look at pedal triangle DEF when pedal point P and the centroid of triangle ABC coincide.

Is there anything special about the pedal triangle DEF? Certainly, when the centroid and the pedal point coincide, the pedal triangle will always lie completely within the original triangle (as long as the triangle is acute). Any other findings? What about if the triangle is obtuse?

Next, we will look at pedal triangle DEF when the pedal point P and the circumcenter of triangle ABC coincide.

Now, is there anything special about the pedal triangle DEF? It appears that the pedal triangle is now the medial triangle. Do you agree? Any other findings?

Next, we will look at the pedal triangle DEF when the pedal point P and the orthocenter of triangle ABC coincide.

Observations about pedal triangle DEF? It certainly appears to be the orthic triangle of triangle ABC. Anything else?