First, let's talk about the definition of a pedal triangle. A pedal triangle is created by the intersections of perpendicular lines from a point in the plane to the sides of a given triangle. Here is an example of one...
The pedal triangle in the above picture is orange. The given triangle is blue and the given point is P.
I began exploring pedal triangles by making a script in Geometer's Sketchpad that created pedal triangles. I used this script to see what happens when we place P in different places. (To try the script for yourself in a GSP file, click here.)
First, I looked to see what happens when P is on one of the sides of triangle ABC. Using GSP, I made the following conjecture: the angle in the pedal triangle at P is equal to the sum of the two angles in the given triangle that are on the same side as P. Look at the picture below for clarification.
Here is a proof of my conjecture:
Here P is on side BC and AB so P is the intersection of the perpendicular from P to the triangle. The only side that it does make sense to look at is side AC. Thus, we get a straight line from P to E that is perpendicular to AC when we look for the pedal triangle when P is at one of the vertices of the original triangle.
In this case, the pedal triangle is the same as the orthic triangle. This is because the orthic triangle is made of the intersection points of the altitudes of the original triangle. These are exactly the same points that make the pedal triangle. Here is a picture of it.
After doing some experimenting, I discovered that the pedal triangle in this case is the same as the medial triangle. This is because the circumcenter is created by the perpendicular bisectors of each side. That means the intersection of the perpendicular from P to a side is at the midpoint. The definition of a medial triangle is the triangle that is created by the midpoints of the sides of the original triangle.
The incenter is created by the intersection of the three angle bisectors. That means the incenter is equidistant from the sides of the original triangle. Because of the special characteristics of the incenter, we can create an incircle that is tangent to the triangle's sides. The pedal triangle, in this case, could also be constructed by the points where the incircle intersects the sides of the triangle. The distance from the incenter to the sides has to be the same and the way we measure distance is along a perpendicular! This means that P is the circumcenter to our pedal triangle. Pretty cool, huh? Here is a picture of the pedal triangle and the incircle.
The centroid is a tough one. It is constructed by the intersection of the medians of the triangle. The centroid is not equidistant from the vertices of sides of the original triangle. It also is not constructed using perpendiculars, like the orthocenter. The centroid's claim to fame is that it splits the median into two parts where one part is 1/3 of the length and the other is 2/3 the length of the median. Could this be influencing the pedal triangle? I couldn't seem to find a special property when P was the centroid. Maybe you'll be able to figure it out. Click here to open a GSP file with this problem already constructed in it.
I found these explorations very intriguing. My knowledge of the triangle centers concepts and my understanding of them were tested and strengthened through this process. Not only do I understand what a pedal triangle is, I also feel much more confident in my understanding of the triangle centers. I recommend this activity to geometry teachers.