What would happen if the pedal point was the centroid of a triangle?
My first observation is that the centroid stays inside the pedal triangle. Consider the following generic picture:
Notice, however, that the pedal triangle may stick out of the original triangle :
From the two pictures above and playing around with this in geometers sketchpad (you can try yourself here) it appears that the triangles are either both acute or both obtuse. This would lead me to conjecture that either the triangles are similar or there is some other generic relationship besides similarity.