PEDAL TRIANGLE
Assignment 9
By Gloria L. Jones
In this assignment the
pedal triangles will be explored based on various locations of the pedal point
P. If P is any point in the plane,
then the triangle formed by constructing perpendiculars to the sides of a given
ABC is called the pedal triangle.
Perpendiculars may also be constructed along the extension of a side if
necessary.
First, we look at the
construction of a pedal triangle given that P is the centroid of the triangle:
Click
here to manipulate a sketch for
various types of triangles with the centroid as the pedal point. It appears that the pedal triangle
never falls completely outside the original triangle. For the acute and right triangles the pedal triangle falls
inside the original triangle but for obtuse it is partly inside the original
triangle and partly outside.
Next we will explore the
pedal triangle given that P is the incenter of the triangle:
Click here to manipulate a sketch for various types of
triangles using the incenter as the pedal point P. It appears that the pedal triangle will always be an acute
triangle in this case and will always remain inside the triangle.
If P is the orthocenter of
a triangle, we have the following sketch:
Click
here to manipulate a sketch where
pedal point P is the orthocenter.
When P is in the interior of the triangle the pedal triangle remains
inside the figure. As point P
exits from a vertex, the pedal triangle becomes degenerate and as P moves
outside the triangle the pedal triangle is located partly inside and partly out
the figure
Through this exploration
it has been discovered that degenerate pedal triangles are not limited to cases
only when P is one of the vertices of the triangle.
Next we will explore the
pedal triangle given that P is the circumcenter of the triangle:
Click
here to manipulate a sketch where
P is the circumcenter of a triangle.
It appears that when P is inside the triangle the pedal triangle is
acute, when P is on the side of the triangle the pedal triangle is right and
when the P is outside of the triangle the pedal triangle is obtuse. In all cases, however, the pedal triangle
remains inside the original triangle.
Finally we will explore an
animation of pedal point P about the in circle of DABC.
Traces of the midpoints of the pedal triangle will then be examined:
