Let triangle ABC be any triangle. Then if P is any point in the plane, then the triangle formed by constructing perpendiculars to the sides of ABC (extended if necessary) locate three points R, S, and T that are the intersections. Triangle RST is the Pedal Triangle for Pedal Point P.
Click here for a GSP script for the general construction of a pedal triangle to triangle ABC where P is any point in the plane ABC.
Let's suppose that pedal point P is the centroid of triangle ABC. The following is observed:
Two vertices are always on one side of triangle ABC.
Here are a few examples.
When triangle ABC is an equilateral triangle, the pedal triangle is the medial triangle of triangle ABC.
When triangle ABC is an isoceles triangle so is the pedal triangle.
Let the incenter of triangle ABC be the pedal point.
Two observations can be made. First, the pedal triangle is always inside triangle ABC. Click here to try to find a counter example.
Second, just like the centroid, when triangle ABC is an isoscoles triangle so is the pedal triangle. Click here for a GSP sketch.
What if the pedal point is the orthocenter of triangle ABC?
When a point is an orthocenter of a triangle it will either be inside the triangle or outside. What we notice is that the pedal triangle does the same as what the orthocenter or pedal point does.
So why does this happen?
The pedal triangle is constructed by taking the perpendiculars from point P to the sides of the triangle. Where those lines intersect is where the vertices of the pedal triangle are. The orthocenter is the intersection point of the three altitudes of the triangle. An altitude is the perpendicular to a side from the opposite vertex. So the lines that are perpendicular from the pedal point P are the lines in which the altitudes are contained. Thus if the altitude goes outside of the triangle (which is the case when the orthocenter is outside of the triangle) then the pedal triangle will be outside of the triangle.
The circumcenter as the pedal point, I believe makes the most interesting observations.
First, the pedal triangle is ALWAYS the medial triangle of triangle ABC. I first noticed this by observation. I then measured the distances and found that the vertices of the pedal triangle were in fact the midpoint of the segments that formed triangle ABC. Click here for the GSP sketch that shows this. The reason for this is that the circumcenter equidistant from each side of the triangle. That means that it lies on the perpendicular bisector of each side. So when one drops the perpendicular to each side of the triangle from the circumcenter, it intersects the triangle at the midpoints. Thus forming the medial triangle.
Second, when the circumcenter or pedal point lies outside of triangle ABC, it passes through a vertex of the pedal point to go on the outside. Click here to try it out on the GSP sketch. Unfortunately I am not sure why this is true at this time.
When the pedal point is on a side of the original triangle ABC, then it acts as a vertex of the pedal triangle. This is best illustrated by the GSP animated sketch.
When the pedal point is one of the vertices of the triangle then a straight line is formed. It is actually the altitude from that vertex to the opposite side.