Assignment # 9

Pedal Triangle

By: Elizabeth Gore

In this investigation, we will be exploring the Pedal Triangle. You may ask, What the heck is a pedal triangle?

Well it is simple.

The pedal triangle is the triangle formed when the perpendiculars are constructed from an arbitrary point P

to the extension of each side of a triangle

Below is a sketch of pedal triangle RST to given triangle ABC.

Triangle ABC will be the triangle we use throughout this exploration.

What are some of the common characteristics of the pedal triangle??

1. When P is the circumcenter of triangle ABC, the pedal triangle RST is also the medial triangle.

2. When P is the orthocenter of triangle ABC, the pedal triangle RST is also the orthic triangle.

3. When P lies on any part of a side of triangle ABC, the pedal point P is a vertex of the pedal triangle RST.

4. When P lies on any vertex of triangle ABC, the pedal triangle RST degenerates to a line that forms an altitude of triangle ABC.

Want to try it yourself?

Click HERE for a GSP sketch to manipulate into the different formations.

Here we have added the construction of the circumcircle to the sketch,

because the pedal triangle also shows some interesting characteristics concerning the circumcircle.

We can see that the vertices of the pedal triangle are collinear when the pedal point P lies on the circumcircle.

The line that is formed is part of the Simpson Line.

To see an animation of P following the path of the circle and maintaining the Simpson Line click HERE.

The envelope that the Simpson Line follows forms as the pedal point moves around the circumcircle is called the DELTOID.

Now let's consider the case when P lies on the incircle of ABC.

When P is the point of intersection of a side of triangle ABC and the incircle, P is a vewrtex for the pedal triangle RST.

As P follows around the incircle, the loci of the midpoints of the sides of the pedal triangle RST form three ellipses. Shown below.

If triangle ABC happens to be a right triangle,

the loci of the midpoints of the sides of pedal triangle RST are 2 ellipses and a circle,

which is tangent to the two perpendicular sides of triangle ABC.

The center of the circle is the midpoint of the segment joining the incenter and the vertex of the right triangle.

We have now constructed an excircle of a new triangle ABC.

I made this given triangle smaller so that the entire excircle could be viewed, but it does not change the properties.

When we trace the loci of the midpoints of the sides of the pedal triangle RST, we observe that three elipses are formed.

The foci of one of the eliposes lies on the bisector of angle BAC.

These points are the center of the excircle and the intersection of the excircle and the angle bisector.

The segments which join the foci of each of te other two ellipses are , themselves, bisected by a bisector of an exterior angle of triangle ABC.

To see the animation of P traveling around the excircle click HERE

This completes our exploration of the Pedal Triangle