Assignment Nine: Pedal Triangles

By: Melissa Wilson

In this assignment we will look at how to construct a pedal triangle and then look at some special cases. To begin the construction of the pedal triangle, make a triangle ABC and select a pedal point, P. P can be ANY point in the plane.

Now we need to construct the perpendicular lines from P to the lines formed by extending each side of triangle ABC. Each of these lines will have intersections at L,M, and N, as shown below.

The intersections of the perpendicular lines with each side (at L, M, and N) will be the vertices for the pedal triangle.

Now, cleaning up some of the displayed lines gives the pedal triangle LMN for the original triangle ABC and pedal point P. Use this GSP file to move P to other locations.

Now we will look at a few interesting places to have pedal point P. First, what if P was the incenter of the triangle?

You can also notice that when P is at the incenter of ABC, then P is also the circumcenter of the pedal triangle that is formed. Recall that the circumcenter is the intersection of the perpendicular bisectors of each side, as shown below in orange.

Next, what if P is at the orthocenter of the triangle? The pedal triangle LMN is the orthic triangle for triangle ABC (vertices of the orthic triangle are located at the feet of the altitudes for each side).

The orthocenter for triangle ABC can be located outside the triangle, as shown below. Then the pedal triangle LMN forms the orthic triangle for triangle ACP.

Next, we will look at if P is the circumcenter of triangle ABC. The pedal triangle LMN formed is the medial triangle for triangle ABC, which is formed by the midpoints on each side.

If the circumcenter is outside triangle ABC, you get the picture shown below. Notice the pedal triangle LMN is still the medial triangle for triangle ABC.