By using Geometer's Sketchpad we can explore
various relationships of Pedal Triangle. Given a point P and a
triangle ABC, a __Pedal Triangle__ is defined as the constructed
triangle whose polygon vertices are the feet of the perpendiculars
from P to the sides of triangle ABC. This point P is called the
__Pedal Point__.

First we are given the Pedal Point P and a triangle ABC.

Next, extend the sides of triangle ABC and find the perpendicular lines from point P to these sides. Label the intersections with points R, S, and T.

Finally, connect these intersection points to form the Pedal Triangle RST.

Click here for a Geometer's
Sketchpad script tool for the Pedal Triangle.

What would happen to the Pedal Triangle RST
if point P were *any* point in the same plane as triangle
ABC? Try and translate point P
to observe the relationships between point P, the Pedal Triangle
RST, and our given triangle ABC.

Varying the Pedal Point P

What will happen to the Pedal Triangle RST if the pedal point P is...

**centroid** of triangle ABC? Watch the Pedal Triangle RST
as point P moves to the centroid.

The Pedal Triangle now has all three vertices on the sides of our original triangle ABC. Regardless of the size or shape of triangle ABC, the centroid will always lie inside of triangle ABC (as long as triangle ABC is not degenerate!). Since this means point P will always lie inside triangle ABC as well, the vertices of triangle RST will remain along the sides of triangle ABC.

**incenter** of triangle ABC? Watch the Pedal Triangle RST
as point P moves to the incenter.

Triangle RST now has all three vertices on the sides of triangle ABC, similar to the centroid example above. Regardless of the size or shape of triangle ABC, the incenter will always lie inside of triangle ABC (as long as triangle ABC is not degenerate!). By experimenting with different sizes and shapes of triangle ABC, we can see that the vertices of triangle RST will remain on the sides.

**orthocenter** of triangle ABC? Watch the Pedal
Triangle RST as point P moves
to the orthocenter.

Triangle RST now has all three vertices on
the sides of triangle ABC, but do you notice something else? Each
of the vertices also lie along the *altitudes *of triangle
ABC. When triangle ABC is acute, these vertices R, S, and T lie
on *both* the sides of triangle ABC, and on the altitudes.
When triangle ABC is obtuse, its orthocenter lies outside the
triangle. Consequently, the vertices of the Pedal Triangle will
still lie on the altitudes, but they are no longer on the sides
of triangle ABC. By experimenting with different sizes and shapes
of triangle ABC, we can see that the vertices of triangle RST
will remain on the altitudes of triangle ABC.

**circumcenter** of triangle ABC? Watch the
Pedal Triangle RST as point P moves
to the circumcenter.

Similar to the orthocenter above, triangle
RST has all three vertices on the sides of triangle ABC. This
time, each of the vertices also lie along the *perpendicular
bisectors *of triangle ABC. When triangle ABC is acute, these
vertices R, S, and T lie on *both* the sides of triangle
ABC, and on the perpendicular bisectors. When triangle ABC is
obtuse, its circumcenter lies outside the triangle. Consequently,
the vertices of the Pedal Triangle will still lie on the perpendicular
bisectors, and they remain on the sides of triangle ABC. By experimenting
with different sizes and shapes of triangle ABC, we can see that
the vertices of triangle RST will remain on the perpendicular
bisectors and the sides of triangle ABC.

**a side** of triangle ABC? Watch the Pedal Triangle
RST as point P moves to a side
of triangle ABC.

When the Pedal Point P moves to one of the
sides of triangle ABC, we can see P actually becomes on the of
vertices of the Pedal Triangle. As shown by the image below, when
point P is on the segment AC, point P and point T become the *same
point*. The remaining vertices of the Pedal Triangle, S and
R, will each lie on the other sides of triangle ABC when it is
acute. If triangle ABC is obtuse, the remaining vertices of triangle
RST can lie outside the triangle ABC. By moving point P to different
points on each side of our triangle ABC, we can see that point
P will be the same point as one of the vertices of triangle RST.

**a vertex** of triangle ABC? Watch the Pedal
Triangle RST as point P moves to
a vertex of triangle ABC.

When the Pedal Point P moves to one vertex
of triangle ABC, we can see all three vertices of triangle RST
are collinear. This represents only one condition in which all
three vertices of the Pedal Triangle RST are collinear. This is
also called a degenerate triangle. This segment SR, shown below,
is known as the **Simson Line**. By viewing the animation,
we can see each time point P is moved to a vertex of triangle
ABC, the vertices of the Pedal triangle RST will be collinear.
By moving point P to different vertices of our triangle ABC, we
can see that point P and all three vertices of triangle RST are
collinear.

We can see in the example above
that when the Pedal Point P is moved to a vertex of triangle ABC,
the vertices RST of our Pedal triangle are collinear. In other
words, our Pedal triangle is reduced down to a single line. This
line is called the **Simson Line**. By experimenting with different
positions of point P in Geometer's Sketchpad, we can see there
are other ways to get the Simson Line.

In fact, there are several points around our triangle ABC where the Pedal triangle is reduced to this line. What relationships do these other points have to each other? What relationships do these points have to our original triangle ABC? Experiment with point P and try to figure it out. Move point P all around triangle ABC and note where P will have to be to get the Simson Line.

What do we already know about the Simson Line?

The Pedal triangle becomes the Simson Line when P is placed at each of the vertices of triangle ABC.

The Pedal triangle becomes the Simson Line when P is placed around our triangle ABC in several places.

What can we conclude?

So in order for the Pedal triangle to become Simson's line, P must hit each of the vertices of triangle ABC,

andseveral points around our triangle ABC. Let's try experimenting with the circumcircle of triangle ABC. By definition, the circumcircle of triangle ABC passes through each vertex, and has the circumcenter as it's center.So, the

circumcircleprovides us with both, the vertices of triangle ABCandseveral points around it! Let's test our conjecture to see if it works for all points on the circumcircle.Constructing the circumcircle of triangle ABC, and move the Pedal point P around that circle. We can see that in fact, for all points on the circumcenter, our Pedal triangle will always become the Simson Line! Watch the Pedal triangle as point P moves around the circumcircle of triangle ABC.

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