Pedal Triangles

By

Brooke Norman

What is a pedal triangle?

We will first construct a triangle and pick any arbitrary point, P, in the plane.

The
next step is to mark the intersections of the perpendicular lines to each side
of ABC from point P. The lines may
need to be extended in order to see their intersections, labeled R, S, T.

We
will now connect the points RST to form the pedal triangle for point P.

To
form your own pedal triangle from any point, you may use this script tool.

Now
lets see what happens when we place P in different places:

-P
is at the centroidÉ

Remember,
the centroid is the common intersection of the
medians.

As
you can see, the vertices of the pedal triangle lie
on the sides of the original triangle.

-P
is at the incenterÉ

Remember,
the incenter is the point on the interior of the
triangle that is equidistance from all three sides.

Just as with the centroid, the vertices of the pedal triangle when
P is the incenter will always lie on the sides of the original triangle.

-P
is at the orthocenterÉ

Remember
that the orthocenter is the common intersection of the three lines containing
the altitudes.

This
is with the orthocenter inside ABC.

This
is with the orthocenter outside of ABC.

As you can see, it appears that as long as the orthocenter remains inside of the triangle, the pedal triangle remains similar to the previous two cases. Take notice that no matter where P is located, the vertices of the pedal triangle remain on the altitudes of the original triangle.

-P is at the circumcenterÉ

Remember that the circumcenter is the point in the plane equidistance
from the three vertices of the triangle.

When
the circumcenter is inside ABC.

When
the circumcenter is outside of ABC:

The case of the circumcenter is similar to the orthocenter.
However, in this case, the vertices of the pedal triangle always lie on the
perpendicular bisectors of the original triangle.

-P
is the center of the nine point circle for triangle ABCÉ

-P
is on a side of ABCÉ

Take
notice that P lies on one of the vertices. Which ever side P is on, the vertex for that side is the
same.

-P
is one of the vertices of ABCÉ

It
appears that there is no pedal triangle for this case. This is a special case and a degenerate
triangle has been formed. The
three vertices of the pedal triangle are collinear. The green line segment above is called the Simson line.

Are
there any other conditions in which the three vertices of the pedal triangle
are collinear, or a degenerate triangle is formed? How about when P lies on the circumcircle. Click here
to see what happens.

As
you can see, anytime P lies on the circumcircle, the simson line is formed.

Return
to BrookeÕs
EMAT 6680 Homepage.