BY: Sharren M. Thomas
I begin by constructing the pedal triangle RST below.
When the pedal point is the same as the Centroid or the incenter then the pedal triangle is located in the interior of the original triangle.
When P corresponds to the orthocenter of a triangle various scenarios occur. Below the pedal triangle dissipates into a line with one of the vertices R at the intersection of the orthocenter and the original triangle. See below
When P is the orthocenter of an equilateral triangle then the pedal triangle is the medial triangle of the larger triangle. See below
When P is the circumcenter of a triangle the following observations follow. When P is the circumcenter of an equilateral triangle, the pedal triangle will be the medial triangle for the larger triangle. See below.
In addition, if the pedal point P is the circumcenter of a triangle that is obtuse or acute, the pedal triangle will still be the medial triangle, see both below. Additionally, it appears that when P is inside the triangle the pedal triangle is acute, when it is on the side of the triangle the pedal triangle is right and when P is outside the triangle the pedal triangle is obtuse. In all cases, the pedal triangle remains inside the original triangle.