For this exploration we will take a look at certain arrangements of the pedal point and pedal triangle in relation to special characteristics of triangles.
In particular, we will examine the following situations:
1) When the pedal point P is the centroid of Triangle ABC.
2) When the pedal point P is the incenter of Triangle ABC.
3) When the pedal point P is the orthocenter of Triangle ABC (inside and outside of the triangle).
4) When the pedal point P is the circumcenter of Triangle ABC (inside and outside of the triangle).
5) When the pedal point P is the center of the nine point triangle of Triangle ABC.
Let’s first review how to construct the pedal triangle from the pedal point for a given triangle ABC.
Let P, the pedal point, be any point in the plane, then the intersection points of the perpendicular lines from P to the sides (extensions) of Triangle ABC form the pedal triangle.
Here is a view of Pedal Triangle DEF from Pedal Point P outside of Triangle ABC:
Here is a view of Pedal Triangle DEF from Pedal Point P inside of Triangle ABC:
Here is the GSP file for the pedal point and pedal triangle to do your own manipulations.
Now that I have a Pedal Point and Pedal Triangle to work with, I can start with the situations.
1) When the pedal point P is the centroid of Triangle ABC.
First, I need to construct the centroid and pedal triangle of Triangle ABC:
There does not seem to be anything too special about the construction, but we do know the pedal triangle will stay inside the Triangle ABC, because the pedal point is the centroid; the centroid is where the medians of the triangle intersect which only occurs inside the triangle.
2) When the pedal point P is the incenter of Triangle ABC.
Here is the incenter and pedal triangle created using the incenter as the pedal point of Triangle ABC:
Again, the pedal triangle will always be inside the Triangle ABC.
The pedal triangle does help form the incircle from the perpendicular lines:
The vertices of the pedal triangle are the points of tangency of the incircle to the interior of Triangle ABC; the intersection point of the points of tangency to the opposite vertices of Triangle ABC is called the Gergonne Point:
Using Ceva’s Theorem, http://jwilson.coe.uga.edu/EMT725/Ceva/Ceva.html, we know the points of tangency are concurrent because:
BD/DA x AE/EC x CF/FB = 1
Here is the GSP file for the Gergonne Point to do your own manipulations.
3) When the pedal point P is the orthocenter of Triangle ABC (inside and outside of the triangle).
Here is the orthocenter and the pedal triangle located inside Triangle ABC:
Notice by definition of the construction of the pedal triangle and the orthic triangle, both using the perpendicular segments from the pedal point/orthocenter to the side of the triangle, they are the same.
This is what happens when the orthocenter is on the outside of Triangle ABC:
Still, the orthic triangle and pedal triangle are the same as the perpendiculars to each side remain the same.
4) When the pedal point P is the circumcenter of Triangle ABC (inside and outside of the triangle).
Here the circumcenter is constructed using the perpendicular bisectors of the sides of the Triangle ABC:
Because the pedal point also uses the same perpendicular lines to create the pedal triangle, the pedal triangle will always remain in Triangle ABC. Here is what happens when the circumcenter/pedal point is outside Triangle ABC:
The pedal triangle is also the medial triangle because the circumcenter also uses the midpoints of the sides; D is the midpoint of AB, E is the midpoint of AC, and F is the midpoint of CB.
5) When the pedal point P is the center of the nine point circle of Triangle ABC.
When the center of the nine point circle lies outside of Triangle ABC, so does part of the pedal triangle:
