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.
Click HERE
to see an animation of 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.
This phenomenon is
better seen with an animation. So click HERE.
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.
For an animated view
click HERE
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
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