Assignment 9:  Pedal Triangles

By:

Jonathan Sabo

For this writeup we will explore what happens to the pedal triangle if the pedal point is the centroid, incenter, circumcenter, or orthocenter of a traingle.

We will start by observing what happens if pedal point P is the centroid of triangle ABC.

We notice that when the triangle is an acute triangle, the pedal triangle remains inside of the original triangle.  This is similar to the actual centroid.  Whenever the triangle is acute the centroid is going to stay inside of the triangle.

When the original triangle is an obtuse triangle, part of the pedal triangle will move outside of the original triangle.  This is happening because the centroid is leaging the triangle, so the pedal triangle is following the centroid out of the triangle.  Both the centroid and the pedal triangle leave through the vertices of the triangle.

Now we will observe what happens if pedal point P is the incenter of triangle ABC

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We can see that if pedal point P is the Incenter of the triangle then the pedal triangle will lie inside of the original triangle.  This will happen for all cases if the original triangle is right, acute or obtuse.  The pedal triangle will never leave the original triangle.

Now we will observe what happens if pedal point P is the circumcenter of triangle ABC.

If the original triangle is an acute triangle we can see that the pedal triangle stays inside of the original triangle.  All of the vertices of the pedal triangle are at the midpoints of the sides of the original triangle and will always stay inside of the original triangle.

If the original triangle is an obtuse triangle the pedal point P will be on the outside of the original triangle.  Whenever the pedal point leaves the original triangle it will pass through the midpoint of the side that it is going through.

Now we will observe if the pedal point P is the Orthocenter of triangle ABC.

We can see that if pedal point P is the Orthocenter it will always lie inside of the original triangle.  If the original triangle is an acute triangle the pedal triangle will always lie inside of the original triangle.  There are some cases where the pedal triangle will be outside of an obtuse triangle.