In this assignment, I look at the cases where the pedal point P of any triangle ABC is located on a side or vertex of the triangle ABC. Cases will be explored where triangle ABC is equilateral, acute, right, and obtuse.

When triangle ABC is equilateral and the pedal point P is located on a side of the triangle, the pedal triangle, RST's three vertices are located on the sides of triangle ABC. When the three vertices of triangle RST are located so that they coincide with the midpoints of triangle ABC, triangle RST is also an equilateral triangle. If point P coincides with any vertices of triangle ABC, then the pedal triangle's vertices become collinear. The pedal triangle becomes a degenerate line when it coincides with the altitudes of triangle ABC.
A GSP sketch that animates this situation is available for you to view.

Here again, the vertices of the pedal triangle are located on the sides of triangle ABC when pedal point P is on a side of triangle ABC. The pedal triangle RST becomes a degenerate line and coincides with the altitudes of triangle ABC if pedal point P is at a vertex of triangle ABC.
A GSP sketch that animates this situation is available for you to view.

In a right triangle, if the pedal point P is located on the hypotenuse, triangle RST's vertices are on the sides of triangle ABC and the vertex on the the hypotenuse is a right angle. Thus, triangle RST is a right triangle when the pedal point P is on the hypotenuse of triangle ABC. Again, triangle RST becomes a degenerate line as it coincides with the altitudes of triangle ABC.
A GSP sketch that animates this situation is available for you to view.

One vertex of the pedal triangle RST is always exterior trianglek ABC when the pedal point P is located on the side of triangle ABC. As expected, triangle RST becomes a degenerate line as it coincides with the altitudes of triangle ABC. This explains why one vertex of triangle RST is located outside triangle ABC since the altitude of an obtuse triangle is exterior.
A GSP sketch that animates this situation is available for you to view.