Assignment 9

Pedal Triangles

By Carly Coffman


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 (extended if
necessary) locate three points R, S, and T that are the intersections. Triangle RST is the
Pedal Triangle for Pedal Point P.

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Observations:



Pedal Point as the Centroid

First, we will explore what happens to the pedal triangle when the pedal point, P, is the centroid. The centroid is the intersection of the midpoints of the sides of a triangle.

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Observations:

 


Pedal Point as the Incenter

Secondly, we will explore what happens to the pedal triangle when the pedal point is the incenter. The incenter is the intersection of the angle bisectors of a triangle.

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Observations:

 


Pedal Point as the Orthocenter

Thirdly, we will explore what happens to the pedal triangle when the pedal point is the orthocenter. The orthocenter is the intersection of the triangle altitudes at the vertices.

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Observations:

Notice that the pedal triangle is the orthic triangle when the pedal point is the orthocenter. Let's look at the definitions of the triangles to see why this is true.

Definitions:

Since the pedal point is the orthocenter, the altitudes of the triangle at the vertices are on the line that is perpendicular to the sides through the orthocenter, or pedal point. Therefore, when the pedal point is the orthocenter, the pedal triangle is the orthic triangle.


Pedal Point as the Circumcenter

Fourthly, we will explore what happens to the pedal triangle when the pedal point is the circumcenter. The circumcenter is the intersection of the perpendicular bisectors of the sides of a triangle.

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Observations:


Pedal Point on the Sides

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Observations:


Exploration

Locate the midpoints of the sides of the Pedal Triangle. Construct a circle with center at the circumcenter of triangle ABC such that the radius is larger than the radius of the circumcircle. Trace the locus of the midpoints of the sides of the Pedal Triangle as the Pedal Point P is animated around the circle you have constructed. What are the three paths?

 

GSP Solution

We get three elliptical loci from the midpoints of the pedal triangle.


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