Stephani Eckelkamp

The Pedal Point as the
Orthocenter

Question: Are there any special properties of the pedal triangle when
the pedal point is located on the orthocenter?

Exploration:

First let us begin with a picture when
the pedal point falls on the orthocenter.

For our first exploration we will
only focus on the original triangle being acute.

Working with this GSP file, see if
you can find any special properties on the pedal triangle.

Try using the script tools to see
if any special properties can be seen.
(If you do find something, make sure that you can move the triangle to
make sure the property holds true for different movements of the construction
(only working with acute triangles).

Did you find anything interesting?

Try creating a perpendicular line
from the pedal triangle point and the side it lies on.

Here is an example.

Line AD is the perpendicular line
created from highlighting A and the side of the original triangle it is located
on.

Do this for all of the angles.

Your picture should look something
like this.

While keeping your original
triangle acute, see if you can find any special properties of the pedal
triangle. Explore segments,
angles, area, and anything else that you think might have a relationship of
interest.

Here is one relationship that I
found interesting.

From the above measurements we can
see that from our construction the perpendicular lines are angle bisectors of
the pedal triangle.

Move the triangle around to see if
our construction supports this idea.

Click here
to move construction.

Did you notice anything different
about the measurements when the triangle was obtuse? How about when it was a right triangle?

Here is a picture when the original
triangle is right.

What is segment AB forming in our
original triangle? Is angle B
still being bisected?

Can you create the altitude for the
other two sides? Click here to work with this file.

What if the original triangle is
obtuse?

Here is a picture:

Here we can see some things
changed. The pedal triangle is no
longer enclosed in the original triangle, only one angles is being bisected by
a perpendicular line, and the orthocenter is not located in either of our
triangles.

Explore the pedal triangle when the
original triangle is obtuse.

What do you notice about any angles
being bisected? Is it possible for
more than one angle to be bisected when the original triangle is obtuse? Why or why not?

Here is another picture:

Further Explorations:

What if the pedal point is on the
incenter?

Circumcenter?

Center of the 9 point circle?

Centroid?

Side of the original triangle?

Blank GSP file with scripts tools to further
explore the above mentioned extensions