**Pedal Power**

**A Peek at Pedal Triangles**

By Rebecca L. Adcock

Definition of a Pedal
Triangle: Let triangle ABC be any triangle. Choose Point P as any point in the
plane. Construct lines perpendicular to
each side of the triangle that pass through P. The triangle formed by the
intersections of the lines with the triangle is the Pedal Triangle.

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If we choose Point P to also
be the centroid of triangle ABC, we get this:

When triangle ABC is
isosceles, the pedal triangle is similar. In the picture below BDE is the Pedal
triangle. Line segment DE is parallel to line segment AC, sides are
proportional and angles are congruent.

As triangle ABC approaches an
equilateral triangle, the pedal triangle becomes identical to it. Below is a
picture of the two almost being identical. Any closer and they will merge into
one.

If we choose Point P to be
the incenter of the triangle, we can get this:

If we choose
the orthocenter as the Pedal Point…….

If we choose to locate the
orthocenter outside of the triangle also have it be the pedal point, the same
thing happens. The pedal triangle and the given triangle are one and the same.
Once again the only intersections occur at the vertices.

If P is the
circumcenter….

Here’s another case where the
pedal triangle is a line segment. The third vertex lies outside the triangle
past the vertex opposite the segment.

Here’s a more ‘normal’
looking pedal triangle.

Let’s try something more
interesting with these pedal triangles. We will locate the midpoints on the
sides of the pedal triangle
and trace the locus of these midpoints as the pedal point moves
on a circle centered at the circumcenter. First we’ll
see what happens when the circle is larger than the circumcenter.

Here’s what the triangles
look like with the traces of the midpoints. The locus are
ellipses and partial ellipses.

To see this in action, click
here.

If we use
the circumcircle as the path…

… the
trace ellipses look a little more complete. To see this in action, click here.

If we use a
path smaller than the circumcircle….

… the
ellipses are complete. To see this is action, click here.