Exploring the Pedal Triangle

by

Rita Meyers

This exploration deals with the construction of
a pedal triangle and investigating different aspects of it.
First given triangle ABC we must construct
the Pedal Triangle RST
Now that we know how to construct a
pedal triangle...what happens when you make the arbitrary point
P equal to the incenter...the point inside of the triangle that
is equidistant from all sides of triangle ABC.
Now we construct the incircle of triangle
ABC...
If you look closely you will see that
the incircle of triangle ABC is also the circumcircle of our pedal
triangle RST

Exploration #1

We also notice that the pedal triangle and the
pedal point never travel outside of the original triangle
What happens when the pedal point is
also the circumcenter?
Just by definition of a pedal triangle
we can prove that the vertices of the pedal triangle are the midpoints
of the sides of our original triangle ABC

Exploration #2

Again the pedal triangle never travels outside
of the original triangle; however, the pedal point will exit triangle
ABC through the vertices of the pedal triangle or the midpoints
of triangle ABC
Now let's look if the pedal point is
the centroid
Again while exploring we can see that
the pedal triangle will never move outside of the original

triangle ABC

Exploration #3

Although the triangle never moves outside triangle
ABC; you will notice that it can change from a triangle to simply
a line segment
Lastly let's look at the pedal point
being the orthocenter
If you look at the exploration we can
see that the pedal triangle does move outside of the triangle
ABC

Exploration #4

We can also see that the pedal point will always
exit the original triangle through the vertices. It also
appears that when the pedal point becomes one of the vertices
of the the original triangle that all three sides of the pedal
triangle merge forming a segment

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