The Pedal Triangle

By: Ginger Rhodes

What is the Pedal Triangle for a Pedal Point P?

Construction of the Pedal Triangle: Begin with ∆ABC. If P is any point in the plane, then the triangle formed by points of intersection of the perpendiculars from point P to the sides of ABC is the Pedal Triangle.

Click HERE for a GSP file that allows you to explore the Pedal Triangle and Pedal Point.

Did you discover anything from your exploration?

First, I will discuss when the Pedal point lies on a side of the ∆ABC. Click HERE for an animation of this exploration.

The Pedal Point P is on the segment AC. Notice the Pedal triangle is inside of ∆ABC and the Pedal point is a vertex of the Pedal triangle.

Now, does anything change at the points A and C?

Yes, the points R, S, and T are collinear. This line segment is called the Simpson Line.

What happens if ∆ABC is obtuse?

Part of the Pedal Triangle is inside of triangle ABC and part of it is outside of triangle ABC.

Do you still get the Simpson Line when the Pedal Point is on points A or C?

Yes, you do!

So, what are the conditions in which the three vertices of the Pedal triangle are collinear?

It is the circumcircle of triangle ABC. Click HERE for an animation and HERE for a proof that R, S, and T are collinear.

To find the envelope of the Simpson line as the Pedal point moves along the circumcircle click HERE.

This is called the Deltoid curve.