Investigating Pedal Triangles

by

Lizzy Shaughnessy

Assignment 9

In this write-up, we will investigate pedal triangles. We will look at special cases including when the pedal point P is different centers of a triangle, including the incenter, circumcenter, orthocenter. We will also look at when the pedal point P is on a side of the triangle and when it is a vertex of the triangle.

First we will start with the construction of a pedal triangle. We need triangle ABC and an arbitrary point P, we will also refer to this as the pedal point.

Now we will construct the line through point P and perpendicular to line AB of triangle ABC. We will repeat this process for each side of triangle ABC. In the picture below, the point of intersection between the perpendicular line and the line created by a side of the triangle are labeled point R, S, and T.

Triangle RST is our pedal triangle.

The above picture is an example of when the pedal point is outside of triangle ABC. Now we will see what the pedal triangle looks like when point P is inside triangle ABC.

Here is a more clear picture of the pedal triangles we have constructed thus far.

You can click here to go to my GSP Script Tool page that has a script to construct the pedal triangle and all of the other centers we will discuss.

Now we will look at the pedal triangle when the pedal point P is a center of the triangle ABC. The first center we will look at is the incenter.

What happens to the pedal triangle when the pedal point P is the incenter of triangle ABC?

As it turns out, if the pedal point P is the incenter of triangle ABC then the pedal point P is also the circumcenter of triangle RST. The circumcircle is in orange in the picture below.

What if the pedal point P is the orthocenter of triangle ABC?

When the pedal point P is the orthocenter of the triangle ABC, the pedal triangle RST is also the orthic triangle. Recall that the orthic triangle is the triangle created by the feet of the altitudes.

What if the orthocenter is outside of the triangle ABC? We will change the dimensions of triangle ABC to consider this case.

When the orthocenter (which is also the pedal point P) is outside the triangle ABC, the pedal triangle RST is again the orthic triangle of triangle ABC.

What if the pedal point P is the circumcenter of triangle ABC?

When the pedal point is the circumcenter and the circumcenter is inside triangle ABC, the pedal triangle (triangle RST) is also the medial triangle of triangle ABC. Recall that the medial triangle is the triangle whose vertices are the midpoint of each side.

What if the circumcenter is outside the triangle ABC?

It turns out that when the pedal point P is the circumcenter and the circumcenter is inside triangle ABC, the pedal triangle (triangle RST) is still the medial triangle of triangle ABC.

Now we will see what happens to the pedal triangle when the pedal point P is on a side of triangle ABC.

Pedal point P on side AC:

Pedal point P on side AB:

Pedal point P on side BC:

So when the pedal point P is on a side of triangle ABC, point P becomes a vertex of the pedal triangle!

Finally, we will conclude by see what the pedal triangle looks like if the pedal point P is a vertex of triangle ABC.

When the pedal point P is on a vertex of triangle ABC, the pedal triangle collapses and is no longer a triangle because it is only defined by two points. This line is known as *Simson's Line*.