I have chosen to look at pedal triangles whose centers are special points of triangles that I have encountered in MATH 7200 this semester. Such points include the circumcenter, orthocenter, centroid and when the center is at a vertex of the triangle. For each case we will have a triangle ABC and the pedal triangle of ABC, triangle XYZ with pedal point, P.
Let's begin with the centroid.
By playing around with the vertices of triangle ABC we observe that the pedal triangle XYZ is always inside triangle ABC. The only exception to this is if the pedal triangle becomes a Simson line. A Simson line will occur when all the vertices of the pedal triangle XYZ are collinear. In otherwords, the triangle XYZ becomes degenerate.
Now let us look at the case of P being the orthocenter:
By playing around with the vertices of triangle ABC we observe that the pedal triangle XYZ can either be inside or outside triangle ABC. The only exception to this is if the pedal triangle becomes a Simson line.
Now let us look at the case of P being the circumcenter:
By playing around with the vertices of triangle ABC we observe that the pedal triangle XYZ is always inside triangle ABC.
By playing around with the vertices of triangle ABC we observe that the pedal triangle XYZ is always a Simson line. This means that XYZ will never be a triangle, but always degenerate.
Here is the GSP file with all of the constructions for these different cases: Pedal Triangles