Pedal Triangles

Allyson Faircloth

Here is a GPS tool for the construction of a pedal triangle.

If we use the tool we will end up with a pedal triangle such as the one pictured to the right.

We will now investigate what happens when point P is on a side of triangle ABC.  We find that when this occurs P is the intersection of the perpendicular line through P and that side.  The point P and I have become the same point. Figure 1 below shows a constructed pedal triangle using the GSP tool and Figure 2 shows when P is on a side of triangle ABC.

                                                                        Figure 1                                                                Figure 2

Now we will look at what happens when P lies at a vertex (Figure 4). Figure 3 shows the pedal triangle for P being placed very close to vertex B.

                                                                      Figure 3                                                                    Figure 4

If P is at a vertex, the perpendicular lines through P and the adjacent sides intersect at that vertex, and the pedal triangle no longer exists.  It is now just a line segment through P and the intersection of the perpendicular line through P and the opposite side of the triangle.  For example in Figure 3 we can see that as P gets closer to vertex B, the pedal triangle seems to be collapsing upon itself.  When P finally reaches vertex B, the pedal triangle has just become a line. 

Interestingly we can find other points outside triangle ABC with the same effect on the pedal triangle.  The pedal triangle also becomes a line at these points.  These points lie on the circumcircle of triangle ABC.  At these points, the three vertices of the pedal triangle become collinear. To see this in action, look at the following GSP file and animate point P by going to "Display", choosing "Show Motion Controller."and clicking the arrow for play: 

Point P on the Circumcircle of Triangle ABC.