Let triangle ABC be any triangle. Then if P is any point in the plane,
then the triangle formed by constructing perpendiculars to the sides of
ABC (extended if necessary) locate three points R, S, and T that are the
intersections. Triangle RST is the Pedal Triangle for Pedal Point P. This
definition is illustrated in **Figure 1**.

Now let us investigate the pedal triangle when pedal point P is located in the following positions:

1.What if pedal point P is the Orthocenter of triangle ABC ? Even if outside ABC?

2.What if . . . P is the Circumcenter . . . ? Even if outside ABC?

3.What if P is one of the Vertices of triangle ABC?

If you have not looked number three above, it is worth looking at because the following discussion will be an extension of that pedal point position and its pedal triangle.

As was seen in number three above, the pedal triangle becomes a degenerate
triangle and forms the Simson line when the pedal point is located at one
of the vertices of triangle ABC. Is there any other points where the pedal
triangle becomes degenerate and a the Simson line is formed besides when
located at one of the vertices of triangle ABC? One may want to try looking
at the pedal triangle when the pedal point is on the circumcircle since
the circumcircle passes through each vertice of triangle ABC. This is illustrated
in **Figure 1** below.

From this image you can see that when the pedal point is located on the
circumcircle, the Simson line is formed. If you think that the location
of the pedal point in **Figure 1** is just another special case of when
the Simson line occurs and you are not yet convinced that a pedal point
on the circumcircle always creates a Simson line, go to **GSP**
for an animation of the pedal point on the circumcircle to see for yourself.

Continuing with this discussion on Simson lines. Is there a point on
the circumcircle for the pedal point that makes the Simson line one of the
sides of the triangle ABC? As can be seen in **Figure 2**, there is indeed
such a point, and by moving the pedal point around one can find the location
where the Simson line is the same segment as as the other two sides of triangle
ABC as well.

Now that it has been found that the Simson line can lie along a side
of triangle ABC, is there any way to predict where the pedal point needs
to be located in order for this to occur? By looking at **Figure 2**,
one can notice that the pedal point, **c**(the center of the circumcircle)
and vertice **C** seem to lie on a common line. After constructing the
line through points **c** and **C**, it becomes clear that when the
pedal point is located on this line the Simson line becomes one of the sides
of triangle ABC. **Figure 3** illustrates this point.

In closing this discussion on pedal points and triangles, we will look
at a few other interesting items that they create. If one was to trace the
midpoints of the sides of the pedal triangle, what would you think the resulting
shapes would look like? Circles? Lines? Parabolas? Ellipses? By looking
at **Figure 4**, you can see that the midpoints of the pedal triangle
trace out the paths of ellipses.

Another interesting consequence that occurs, comes from the intersection
between the Simson line and the line segment connecting the pedal point
**P** and the orthocenter **H** of triangle ABC. The point where these
two line segments intersect turns out to be the midpoint of the line segment
through **P** and **H**. **Figure 5** displays this occurrence.

If one was to trace the lines that formed the segments that make up the
sides of the pedal triangle while the pedal point was moving around the
circumcircle, what would one expect to see? When tracing lines, envelopes
generally form defining some type of shape. What shape would you expect
to form from tracing **P** around the circumcircle, to make the previous
question a little more direct? As may have been expected, the lines forming
the pedal triangle when traced form an envelope that resembles a triangle,
as can be seen in **Figure 6**.

This concludes all of the discussions about pedal points and triangles. However, there are many other interesting items to investigate that are not presented here. If the reader feels so inclined to investigate further, some possible investigations may include:

- Animating the pedal point around a circle centered at the circumcenter but with a larger radius than the circumcircle.
- Animating the pedal point around a circle centered at the circumcenter but with a smaller radius than the circumcircle.
- Animating the pedal point around the incircle. The excircle.