**Pedal
Triangles for different locations of the Pedal Point**

Given a triangle and an arbitrary point, a
pedal triangle can be constructed. One may ask what can we say about the pedal
triangle when the pedal point is located at particular locations relative to
the original triangle. We will try to answer this question for the cases where
the pedal point is located at the orthocenter and the circumcenter.

First lets consider the case when the pedal point is located at the orthocenter. We construct a triangle and its orthocenter (H). We then select an arbitrary pedal point (P) and construct the pedal triangle (in red).

When we drag the pedal point to the
orthocenter, the pedal triangle looks very similar to the orthic triangle for
the original triangle. If we drag P back to its original location, we can
construct the orthic triangle (in blue) and then test this conjecture.

We can then merge P to O, and then try
different shapes of the original triangle.

Click
here for a GSP file to try yourself.

We can see that the pedal triangle does
indeed overlay the orthic triangle when P overlays H. But this is true only
when the original triangle is acute. For any other shape of the original
triangle, H is outside the triangle and the orthic triangle cannot be constructed.

Now we will look at the case when the pedal
point is located at the circumcenter. We construct a triangle and locate the
circumcenter (C). We then select an arbitrary point (P) and then construct the
pedal triangle (again in red).

When we drag P to overlay C, we notice that
the pedal triangle looks very similar to the medial triangle of the original
triangle. Again we can move P back to its original location and then construct
the medial triangle (in green) to test this conjecture.

Again we can merge P to C and then change the
shape of the original triangle to see if the pedal triangle overlays the medial
triangle for any shape.

Click
here for a GSP file to try yourself.

In fact we can see that this is true, even
for right and obtuse triangles.