Pedal Triangles

By: Summer Brown

What is a pedal triangle?

Let triangle ABC be any triangle. Let point P be any point in the plane. Construct the perpendiculars to the sides of triangle ABC through point P. The intersections of the perpendiculars with the sides of triangle ABC form three vertices of a new triangle, called the Pedal Triangle for Pedal Point P.

 

In the example below, the pedal triangle for triangle ABC with peal point P is shown in pink.

 

As P is dragged, the pedal triangle changes. Click here to drag point P in GSP.

Click here to see a script of the construction of a pedal triangle.


In this investigation, I will explore what happens to the pedal triangle if the pedal point P is the centroid, incenter, orthocenter, or circumcenter of triangle ABC.

 

Some questions worth investigating are:

1. When does the pedal triangle remain inside the original triangle?

2. What is the relation between the original triangle and the pedal triangle?


Centroid

Let the pedal point P be the centroid of triangle ABC. In these pictures, the original triangle is in yellow and the pedal triangle in pink.

 

The pedal triangle seems to remain inside triangle ABC when triangle ABC is acute or right. When triangle ABC is obtuse, the pedal triangle does one of two things: it remains completely inside triangle ABC, or one vertex reaches slightly outside of the original triangle.

 

I did not find any relation between the original triangle and the pedal triangle.


Incenter

Let point P be the incenter of triangle ABC.

 

In all cases, whether triangle ABC is acute, right, or obtuse, the pedal triangles always remains inside the triangle.

 

One observation to make is that the incircle of triangle ABC is the circumcircle of the pedal triangle. This is because the pedal point is the incenter. To find the vertices of the pedal triangle, you must construct the perpendiculars from the pedal point to each side of triangle ABC. The segment formed from the pedal point to the foot of the perpendicular are radii of both the incircle of triangle ABC and the circumcircle of the pedal triangle. Thus, since the circles have the same center and radii, they are concurrent.

 

Another observation is made when the angle bisectors (dashed lines) of triangle ABC are drawn in. The angle bisectors intersect each side of the pedal triangle at each respective midpoint. Click here to see a proof and explore with the GSP sketch.

 


Orthocenter

Let point P be the orthocenter H of triangle ABC.

 

In all cases, the pedal triangle is equal to the orthic triangle of triangle ABC.

If triangle ABC is acute, then the pedal triangle remains inside triangle ABC.

 

If triangle ABC is right, then the pedal triangle is a straight line, or a degenerate triangle. The pedal point H lies on the vertex of the right angle.

If triangle ABC is obtuse, then the pedal triangle moves outside of triangle ABC. Also, the pedal point no longer remains inside triangle ABC. As the obtuse angle becomes greater, the pedal point moves farther and farther away from triangle ABC.

 

Another interesting observation is that when triangle ABC is equilateral, the pedal triangle also becomes equilateral. In fact, the pedal triangle will always have an area one-fourth the size of triangle ABC when it is equilateral. Click here for a proof of this observation.

 


Circumcenter

Let point P be the circumcenter of triangle ABC.

 

The pedal triangle will always remain inside triangle ABC, even if P moves outside of triangle ABC.

 

The pedal triangle is equal to the medial triangle of triangle ABC.

In all cases, the ratio of the area of triangle ABC to the pedal triangle is 4 to 1. Also, the ratio of the respective perimeters is 2 to 1.

Triangle ABC is divided into 4 congruent triangles, with the pedal triangle being one of these. Thus, the pedal triangle makes up one-fourth of the area of triangle ABC.

The pedal triangle is similar to triangle ABC with sides having lengths 1/2 the lengths of triangle ABC. Click here to explore with a GSP sketch.

 


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