Let's begin by looking at the construction for a pedal triangle.
We will start with a triangle, and pick any arbitrary point P in the plane.
The next step is to construct perpendiculars to the sides of ABC (extended if necessary) and locate the three points R, S, and T that are the intersections.
Next, we will connected the points RST to form the pedal triangle. This is the pedal triangle for this particular point P.
Click here to use the script tool to form your own pedal triangle, placing P anywhere in the plane.
Now, let's look at P when it is...
1.) the centroid.
Remember, the centroid is the common intersection of the medians. Let's see what the pedal triangle looks like for the centroid.
2.) the incenter.
Recall that the incenter of a triangle is the point on the interior of the triangle that is equidistance from all three sides. Let's see what the pedal triangle looks like for the incenter.
3.) the orthocenter.
Remember that the orthocenter of a triangle is the common intersection of the three lines containing the altitiudes. Let's look at the pedal triangle for the orthocenter when it is inside and outside of the triangle.
4.) the circumcenter.
The circumcenter of a triangle is the point in the plane equidistance from the three vertices of the triangle. Let's look at the pedal triangle for the circumcenter when it is inside and outside of the triangle.
5.) on one of the sides of the triangle.
6.) one of the vertices of the triangle.
Are there any other cases for which the Simson Line exists? What if P lies on the circumcircle? Click here to see the animation of P as it rotates around the circumcircle.
As you can see, if P lies anywhere on the circumcircle of the original triangle, then the pedal triangle formed will always be the Simson line.