**Find all conditions in which the three vertices
of the Pedal triangle are colinear (that is, it is a degenerate
triangle). This line segment is called the Simson Line.**

As we begin to look at the relationship between the simson line and the pedal triangle, let's first define those two terms.

** PEDAL TRIANGLE:** 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.(click
here to see a diagram in GSP, and you can move P around to
see how the Pedal Traingle changes)

__SIMSON LINE:____ __In geometry, if one drops
perpendiculars from a point P on the circumcircle of a triangle
ABC to the sides (or their continuations) of the triangle, then
the feet of the perpendiculars turn out to lie on a line, called
the Simson line (of P for the triangle ABC - SEE DIAGRAM BELOW
FOR ILLUSTRATION OF SIMSON LINE)

Now let's explore our Pedal Triangle and see
if we can find when the three vertices of our Pedal Triangle are
colinear, another words forms a degenerate triangle, and when
this is equal to the simson line. The __definition of the Simson
line is a big clue__, let's see if it is what we think..........

First let's take a look at P when it is inside the triangle:

Now that we know that our Pedal Triangle is colinear and and forms the Simson line at each vertices (click here to explore this finding) we must see if there are other cases for P where this is also true. We have already checked for P inside the triangle, so now let's investigate for our point P outside the triangle......

Now let's see if we can find if there exists any point P outside our triangle, where P would again create a Pedal Triangle where the vertices would be colinear, forming a degenerate triangle, and that line would be equal to our Simson Line.

Let's see if we can use what we know about our Simson Line and where we have found our points P where the vertices of our Pedal Triangle are colinear, form a degenerate triangle, and also form the Simson Line?

As I mentioned earlier our big clue in finding our answer involves the definition of the Simson Line, it states that it is formed by the perpindiculars from a point P that is ON THE CIRCUMCIRCLE of the triangle. So let's take the points we have found in our explorations above and see if they all fall on the circumcircle.

We see in our diagram above that our earlier
explorations of different points P (P1, P2, P3, P4, P5) do all
fall on the circumcircle, if we couple this with the definition
of the Simson Line we can answer our initial question. **Find
all conditions in which the three vertices of the Pedal triangle
are colinear (that is, it is a degenerate triangle)**........we
can conclude that for any point P on the Circumcircle, when we
construct the Pedal Triangle from this point P, the vertices of
the Pedal Triangle will be colinear, forming a degenerate triangle,
and the line formed by the vertices are the Simson Line.