Pedal Triangles

By

Brandon Samples

In this exploration, we are going to look at Pedal Triangles. Let's begin with a picture of a generic pedal triangle.

Here's the idea: Take a generic triangle ABC and pick some point P. Construct the perpendicular bisector to the sides of the triangles (extended if necessary) to P. Label the intersection points of the lines through the sides of the triangle with the perpendicular bisectors R, S, and T.

Would you like to try this out for yourself? Here is a script tool: Pedal Triangle

Now, we turn to our exploration of pedal triangles. Since you form a pedal triangle from a given triangle, we can ask the following: What if we take the pedal triangle of a pedal triangle? Will there be any connections? What if we iterate this three times? Let's look at the next figure.

If we take a pedal triangle of a pedal triangle of a pedal triangle of a point and look at the various angles of the original triangle and the last pedal triangle, we see that the angles are the same. Maybe this depends on our configuration. Let's look at another configuration.

So it appears that this behavior happens always!

Now, let's explore what happens when P is the centroid.

After moving the triangle around, we might wonder if we can get the pedal triangle outside of the original triangle. Would you like to try?

Here is a GSP file: Centroid - Pedal Game

Actually it's possible to get the pedal triangle outside. We need to force one of the angles to be largly obtuse. See the following figure:

Now, let's explore what happens when P is the orthocenter.

So as before, we can get the pedal triangle outside of the original triangle, but they will always intesect in a region with positive area unless the triangle degenerates.