Pedal Triangles for different locations of the Pedal Point

Given a triangle and an arbitrary point, a pedal triangle can be constructed. One may ask what can we say about the pedal triangle when the pedal point is located at particular locations relative to the original triangle. We will try to answer this question for the cases where the pedal point is located at the orthocenter and the circumcenter.

First lets consider the case when the pedal point is located at the orthocenter. We construct a triangle and its orthocenter (H). We then select an arbitrary pedal point (P) and construct the pedal triangle (in red).

When we drag the pedal point to the orthocenter, the pedal triangle looks very similar to the orthic triangle for the original triangle. If we drag P back to its original location, we can construct the orthic triangle (in blue) and then test this conjecture.

We can then merge P to O, and then try different shapes of the original triangle.

Click here for a GSP file to try yourself.

We can see that the pedal triangle does indeed overlay the orthic triangle when P overlays H. But this is true only when the original triangle is acute. For any other shape of the original triangle, H is outside the triangle and the orthic triangle cannot be constructed.

Now we will look at the case when the pedal point is located at the circumcenter. We construct a triangle and locate the circumcenter (C). We then select an arbitrary point (P) and then construct the pedal triangle (again in red).

When we drag P to overlay C, we notice that the pedal triangle looks very similar to the medial triangle of the original triangle. Again we can move P back to its original location and then construct the medial triangle (in green) to test this conjecture.

Again we can merge P to C and then change the shape of the original triangle to see if the pedal triangle overlays the medial triangle for any shape.

Click here for a GSP file to try yourself.

In fact we can see that this is true, even for right and obtuse triangles.