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.

- P is the circumcenter of triangle ABD, - click here
- P is the incenter of triangle ABD, - click here
- P is the centroid of triangle ABD - click here
- P is the orthocenter of triangle ABD - click here

__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.

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.

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.

To see how the point is animated click here.

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.

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