I chose the three vertices of the original triangle (in red) and the
point P (the Pedal Point). The script forms the Pedal Triangle (in blue)
by constructing the perpendiculars.
Do special things happen when the pedal point P is a particular point that
is related to the original triangle ABC?
What is the pedal point P is the centroid of triangle ABC?
The blue triangle is the pedal triangle. When point P is the centroid,
the pedal triangle lies completely inside the original triangle.
What if the pedal point P is the orthocenter of the triangle ABC?
When the pedal point P is the orthocenter, we have several cases. The
first case is when the original triangle is an acute triangle. The pedal
triangle lies inside the original triangle. The second case is when the
original triangle is an obtuse triangle. The pedal triangle lies outside
the original triangle. The final case is when ABC is the right triangle.
In this case, the pedal triangle is a degenerate case.
What if the pedal point P is the circumcenter of the triangle ABC?
When triangle ABC is obtuse, the circumcenter/pedal point is outside
the boundaries of the triangle and the pedal triangle is inside. When the
triangle ABC is acute, the circumcenter/pedal point is inside the boundaries
of the triangle and the pedal triangle is inside again.
What if the pedal point P is the center of the nine point circle of the
triangle ABC?
When the pedal point is the center of the nine point circle, the pedal
triangle always has at least one vertex on the original triangle.
What if the pedal point P is on the side of the triangle ABC?
If the pedal point is a point on the side of the triangle ABC, then at
least two of the vertices of the pedal triangle lie on the sides of ABC.
What if the pedal point P is one of the vertices of the triangle ABC?
If the pedal point is one of the vertices of the original triangle ABC,
then the pedal triangle is a degenerate case. It is an altitude of the
triangle ABC.