Assignment 9

Pedal Triangles

by Rives Poe

In this exploration, we are going to observe how the Pedal Triangle changes depending upon the pedal point's location. Sit back and get ready to observe!

First of all, I constructed triangle ABC. From triangle ABC, I put an arbitrary point on the page and constructed perpendicular lines to the sides of ABC. The three points where the perpendiculars cross the triangle is the pedal triangle RST. Here is the construction below:

If pedal point P is the centroid of triangle A, the pedal triangle will look like this:

If the pedal point P is the Incenter:

If pedal point P is the Orthocenter:

The orthocenter is the same for the pedal triangle and triangle ABC. What if the orthocenter is outside triangle ABC? What do you think?

What if the pedal triangle is the Circumcenter????

If the circumcenter is inside the triangle, then the pedal triangle is also inside the triangle and it shares the same circumcenter. Let's see what happens when the orthocenter is outside the triangle:

The pedal triangle divides triangle ABC into 4 congruent triangles.

Let's see what happens when the pedal point falls on the center of the nine point circle.

What if P is on a side of the triangle?

If P is on the side of triangle ABC, it will always lie on a vertex of the pedal point triangle depending on which side you choose!

What if P is on a vertex of triangle ABC?

If P is put on a vertex of triangle ABC, the pedal triangle forms the angle bisector of that vertex.