Assignment 9: PEDAL TRIANGLES

By Nikhat Parveen, UGA

Let triangle ABC be any triangle. Then if P is any point in the plane, then the triangle formed by constructing perpendiculars to the sides of ABC  by locating three points R, S, and T that are the intersections. Triangle RST is the Pedal Triangle for Pedal Point P as shown in the figure below.

 

For a script tool please click HERE.

 

Where would the Pedal triangle be if we locate the Pedal point P inside the triangle.

 

As you notice the figure above, the pedal triangle lies inside the triangle if the pedal point P lies inside the triangle.

Let's see where will the Pedal triangle be if the pedal point P is on one of the side of the triangle.

 

 

The pedal triangle still lies within the triangle when the pedal point is on the side of the triangle.

When the pedal point lies outside the triangle such that the intersections of the altitudes from the pedal point P to the triangles all lies outside the triangle then the pedal triangle also lies outside the triangle and do not intersect any of the sides of the triangle.

 

Click here for the GSP construction of Pedal triangle.

Explorations:

Now let's look at the special cases when the pedal point P is not a random point in a plane but lies at the special points of a triangle, such as centroid, orthocenter, incenter, circumcenter etc.,

Case 1: Centroid

Case 2: Incenter

Case 3: Orthocenter

Case 4: Circumcenter

Case 5:Nine point center

 

Case 1: Centroid

Click here for the GSP exploration.

When the pedal point P is a centroid of a triangle, then the pedal triangle formed lies in the interior of a triangle. Since the centroid of a triangle always lies in the interior of a triangle therefore the pedal triangle of a pedal point P as a centroid will always remain in the interior of  a triangle and will never lie outside the triangle.

 

Case 2: Incenter

Click here for the GSP exploration

When the pedal point P is the incenter of the triangle ABC , we can see from the figure above that the pedal triangle formed lies within the triangle ABC and will remain inside the triangle because the incenter of a triangle always lies with in the triangle. So, regardless of the type of triangle as long as the pedal point P is incenter the pedal triangle will lie interior to the triangle.

Case 3: Orthocenter

 

Click here to explore the location of Pedal triangle by changing the triangle.

When the pedal point P is the orthocenter of the triangle ABC, the pedal triangle lies within the triangle ABC. The location changes when the triangle is changed.

Case 4:Circumcenter

Click here to see the GSP file

Now, when the pedal point P is the circumcenter of the triangle ABC, the Pedal triangle lies interior to the triangle ABC and as the circumcenter of a triangle lies inside the triangle therefore the Pedal triangle will always lies within the triangle as long as the pedal point P lies at the circumcenter of a triangle.

 

Case 5: Nine point center

Click here to explore the GSP file

When the pedal point P is the nine point center, the pedal triangle lies inside the triangle ABC as shown in the figure above. The pedal triangle can lie outside the triangle if the triangle is obtuse and in one of the obtuse triangle the Pedal triangle formed is a degenerate triangle as shown below. In a right triangle, the pedal triangle lies inside the triangle as shown below.

 

          

 

For further exploration of the special cases of Pedal point P, please click on Page 2.

 

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