**Investigations
with the Pedal**

**Triangle**

**by**

**Kimberly
Burrell**

**The pedal triangle, RST, is formed by
choosing any arbitrary point and any arbitrary triangle. The
arbitrary point P is known as the pedal point. In this
investigation the arbitrary triangle will be triangle ABC. The
pedal triangle is formed from the perpendiculars to the sides of
triangle ABC to locate the three intersection points, R, S, and
T. The following is an example of a GSP sketch of a pedal
triangle.**

**In the following we will look at various
pedal triangles that have been constructed by allowing the pedal
point to be specific points related to triangle ABC. We will also
explore other characteristics of the pedal triangle by use of
animation with GSP.**

**Click here**** for a GSP
script of the pedal triangle construction.**

**Now let's allow the pedal point to be located at the
incenter, the point inside of the triangle that is equidistant
from all the sides of triangle ABC. **

**The pedal triangle is located in the incircle
of triangle ABC. The incircle goes through the points of
intersection of the lines through the incenter and perpendicular
to each side of triangle ABC. As one can observe, the incircle of
triangle ABC is actually the circumcircle of the pedal triangle
RST.**

**Now, lets allow the pedal point to be the circumcenter, the
intersection point of the perpendicular bisectors of triangle
ABC.**

**With the circumcenter being the pedal
point, the midpoints of triangle ABC become the vertices of the
peadal triangle.**

**When you allow the pedal point to be the
centroid, the intersection of the medians of triangle ABC, where
will the pedal triangle be located? Let's see ....**

**The pedal triangle will never lie outside
of triangle ABC when the pedal point is the centroid of a
triangle.**

**Let the orthocenter, the intersection of the
altitudes of triangle ABC, be the pedal point.**

**It happens to be that when the orthocenter
becomes one of the vertices of the original triangle, two of the
vertices of the pedal triangle collide into the orthocenter,
giving a degenerate triangle or segment as the pedal triangle. To
investigate this situation with Geometer's Sketch Pad, please ****click
here****.**

**Animations**

**What type of objects do you think will be formed when we trace the midpoints of the legs of the pedal triangle as the pedal point, P, follows the point of the circumcircle of triangle ABC?****Click here****for a Geometer's Sketch Pad animation demonstrating this.****What if we trace the midpoints of the legs of the pedal triangle again, however P follow the path of the circle centered at the circumcenter but a radius larger that of the circumcircle? Can you predict the types of figures that are being traced?****Click here****to see the results.**