Investigation of Pedal Triangles

by Kristy Hawkins

When we investigate triangles, there is a very interesting triangle named the Pedal Triangle.

What is the Pedal Triangle?

Let triangle ABC be any triangle. Then if P is any point in the plane, the triangle formed by constructing the intersections of the lines through P that are perpendicular to the sides of our triangle and the segments of triangle ABC is called the pedal triangle.

Let's investigate this triangle!!

Since this pedal point can be anywhere, what would happen if it is at the incenter of that triangle?

Recall: The incenter of any given triangle is the intersection point of the angle bisectors. This means that it is equidistant from all three sides of the triangle and therefore is the center of the incircle of the triangle.

Look at this. If we construct the incircle, it is the circumcircle of the pedal triangle. Is this true for every triangle? Click here to play with the GSP sketch.

Does this make sense? We know that we find the incenter with the angle bisectors of the original triangle. The radius of the incircle is then found by dropping a perpendicular line to one of the sides of the triangle since this is the closest point from the incenter to the triangle. But that also happens to be the same point that is a vertex of the pedal triangle.

Since this seems to be true, we can make a conjecture that when the pedal point of the blue triangle is at it's incenter, then that same point is also the circumcenter of the pedal triangle. In other words, the pedal point becomes the circumcenter of the pedal triangle when it is the incenter of the original triangle. Wow!

Now what happens when the pedal point is moved to the circumcenter?

Recall: The circumcenter of a triangle is the point that lies on the perpendicular bisector of each segment of that triangle.

This pedal triangle looks familiar! It is the medial triangle! As the pedal point approaches the circumcenter, the vertices approach the midpoint of each side of the given triangle because of the fact that the circumcenter is the intersection of the perpendicular bisectors of the sides of the triangle. This medial triangle is similar to the original triangle, and also one fourth of its area.

If we look at the altitudes of the pedal triangle, or medial triangle, we see that they look to intersect at our pedal point which is also the circumcenter of the given triangle. This would make the pedal point also the orthocenter of the pedal triangle since the orthocenter is the intersection of the altitudes of a triangle.

Does anything interesting happen when the pedal point is moved to the orthocenter?

Recall: The orthocenter of a triangle is found at the intersection of the three lines that contain the altitudes of that triangle.

The pedal triangle is now the orthic triangle as long as the original triangle is acute. P also seems to be the incenter of the pedal triangle while it is the orthocenter of the given triangle.

Didn't we begin with the pedal point being the incenter of the given triangle? So we see that the incenter of a given triangle is the circumcenter of the pedal triangle, the orthocenter of it's pedal triangle, and the incenter of the pedal triangle of the pedal triangle of the pedal triangle. That's an interesting connection. Here is an illustration of it!

In this picture, P is the incenter of the blue triangle, the circumcenter of the green triangle, the orthocenter of the red triangle, and then the incenter of the orange triangle. The bigger pink circle represents the incircle of the blue triangle as well as the cirrcumcircle of the green triangle. The smaller pink circle shows the incircle of the orange triangle. This figure shows us that when the pedal point of the blue triangle is at it's incenter, it is also at the incenter of the third pedal triangle, meaning the pedal triangle of the pedal triangle of the pedal triangle of the original triangle. We can conclude that when P is in this specific position, it will be the incenter of every third pedal triangle, the orthocenter of every second pedal triangle, and the circumcircle of every first pedal triangle. As we continue the pedal triangles of pedal triangles, the cyclic pattern should continue forever.

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