Lisa Brock & Scott Burrell

Assignment 9

Investigations With Pedal Triangles

We begin this investigation by constructing triangle ABC and its pedal triangle for arbitrary point P.  The pedal triangle is formed by connecting the intersections of the perpendicular lines from P to the each of the sides of the triangle.

We will now look at what happens when P is the centroid, G, of triangle ABC.

The pedal triangle is completely inside triangle ABC and appears equilateral.

We will now look at what happens when P is the incenter, I, of triangle ABC.

The pedal triangle remains inside of triangle ABC but no longer appears to be equilateral.

We will now look at what happens when P is the orthocenter, H, of triangle ABC.

The pedal triangle is the orthic triangle.  This makes sense because the orthocenter is the intersection of the altitudes of triangle ABC.  The three points used to form the orthic triangle are feet of the altitudes.  So the vertices of the orthic triangle are formed from the perpendicular from H to each side.  Since P is located at H in this case, the vertices of the pedal triangles are the same as the vertices of the orthic triangle.

Now we well look at whether this holds if the orthocenter, H, is outside of triangle ABC.

This relationship holds when the orthocenter is outside of triangle ABC.

We will now look at what happens when P is the circumcenter, C, of triangle ABC.

The pedal triangle is the medial triangle.  This makes sense because the vertices of the medial triangle are the midpoints of the sides of triangle ABC.  The circumcenter is perpendicular to each side through its midpoint.  Therefore, when P is the circumcenter, the vertices of the pedal triangle would be the midpoints of the sides of triangle ABC.

Now we will see if this relationship holds when the circumcenter is outside of triangle ABC.

The relationship holds when the circumcenter is outside of the triangle.

Now we will look at what happens when P is on one of the sides of triangle ABC.

P becomes one of the vertices of the pedal triangle.

Now we will look at what happens when P is vertex, B, of triangle ABC.

When P is one of the vertices of triangle ABC, the pedal triangle is degenerate (it collapses).

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