Assignment 9

Explorations with Pedal Triangle

by

Behnaz Rouhani

In this assignment the following questions will be answered:

1. What is a Pedal triangle? Given triangle ABD, how could we construct the Pedal Triangle?
2. What are the loci of vertices of Pedal Triangle when P, the Pedal point is a special point (cicumcenter, incenter, centroid, orthocenter).
3. What happens when P animates around incircle or excircle of a given triangle, and midpoints of sides of the Pedal triangle are traced?
4. What happens when P animates around the circumcircle of a given circle, and midpoints of the sides of the Pedal triangle are traced?

1. What is a Pedal triangle? Given triangle ABD, how could we construct the Pedal Triangle?

Given triangle ABD and any point P in the plane, the triangle formed by perpendiculars to the sides of ABD is called a Pedal Triangle for pedal point P.

The Pedal triangle is formed by connecting the intersections of the perpendiculars to the sides of the triangle. Here is the Pedal triangle RST.

What are the loci of vertices of Pedal Triangle when P, the Pedal point is a special point ( cicumcenter, incenter, centroid, orthocenter).

What happens when P animates around incircle or excircle of a given triangle, and midpoints of sides of the Pedal triangle are traced?

Animation around incircle

First,  we will trace the midpoints of the sides of the pedal triangle, then will animate point P about incircle. Remember to merge point P to the incircle as it is not on the circle. As we see, the loci are three ellipses. Here is a snapshot of the animation.

If ABD is a right triangle, then would the loci of the midpoints of the sides of the Pedal triangle as Pedal point P animate about the incircle, look any different? To see it for yourself click here.

Animation around excircle

Now, let's animate point P around the excircle of triangle ABD. To do so we will first trace the midpoints of the sides of the Pedal triangle. Then, animate P around the excircle. Here is the snapshot of the animation, notice that the loci are three ellipses as with the incircle case.

Special investigation

Construct lines (not segments) on the sides of the Pedal triangle, and then trace these lines as the Pedal point P is animated around the excircle. Click here to see the animation.

What happens when P animates around the circumcircle of a given circle, and midpoints of the sides of the Pedal triangle are traced?

First, we will trace the midpoints of the sides of the Pedal triangle, then merge P to the circumcircle. Finally, animate P around the circumcircle. The loci of the midpoints of the Pedal Triangle are three ellipses.  Do you notice anything about the Pedal Triangle? We will talk about this later.

Now,  what if P is animated around a constructed circle with center at circumcenter and radius larger than the radius of circumcircle. The loci of the midpoints of the Pedal Triangle are three ellipses.

Finally, we will construct a circle with center at C and radius less than the radius of circumcircle. The loci of the midpoints of the Pedal Triangle are three ellipses again.

Observation: Loci of the midpoints of the Pedal triangle are three ellipses regardless of whether P is on circumcircle, or on a circle with larger or smaller radius than the circumcircle radius. The main point is that all these paths are centered at the circumcenter, C.

Remember the question that was asked earlier! Here it is.  What happens when P is on the circumcircle of triangle ABD? This choice of P causes vertices of Pedal triangle to be collinear (that means it is a degenerate triangle). In other word Pedal triangle has become a segment. This line segment is called Simson Line.

The envelope that Simson Line follows as Pedal point moves around circumcircle is called Deltoid. To do so, first we need to trace the simson line, and then animate P around the circumcircle.

The Pedal triangle investigation is rich with problems, and this assignment only touched upon few of those cases. Hopefully this will be a good start for future studies.