Lets begin by showing the construction of a pedal triangle for a given triangle.
If P is one of the vertices of the triangle, the following occurs:
tha is the triangle collapses into a line which is an altitude of the original triangle.
If P is on a side of the triangle the following occurs:
that is if p is along one of the sides of the triangle, it corresponds to the point of intersection with the perpendicular from that side.
Lets look and see what happens if the pedal point corresponds to the centroid of a triangle:
If the pedal point is the incenter of the triangle then:
,the pedal triangle becomes the the triangle with a circumscribed circle that is inscribed in the origial triangle.