** Write-Up #9**:

Definition of a Petal Triangle: Let triangle
ABC be any triangle with point P in the plane. Then the triangle
formed by constructing perpendiculars to the sides of ABC (extended
lines if necessary) locate three points R, S, and T that are the
intersections. Triangle RST is the **Pedal Triangle** as shown
below.

For a Pedal Triangle ** script**,
select three points to define a triangle and an outside point.

If pedal point P is collocated with the __centroid
K__ of the given triangle (ABC), then the pedal triangle is
interior to the given triangle. Note that the upper side of the
pedal triangle (red) is not parallel to the base of the given
triangle (black).

If pedal point P is collocated with the __incenter,
I__ of a triangle, then the pedal triangle is also interior
to the given triangle.

If pedal point P is collocated with the __orthocenter,
H__ of a triangle, then the pedal triangle is interior to the
given triangle. Note that the orthocenter is constructed the same
way (note the construction lines) as the pedal triangle when the
pedal triangle is inside the given triangle.

When, due to construction, the orthocenter is located outside the triangle, and the pedal point is collocated with the othocenter, then the pedal triangle exceeds the bounds of the original triangle, but the triangle is consistent with the othocenter, as expected.

If pedal point P is collocated with the __incenter
C__, then the pedal triangle will locate inside the triangle
with sides parallel to the original triangle because the vertices
coincide with the midpoints of the original triangle.

However, if the pedal point P __traces the
circumcircle__ of the original triangle, then the pedal triangle
is reduced to a line called the Simson Line. (Click **here**
for animation of the trace about the circumcircle.) Of course
if the point P is located at a vertex of the original triangle,
the pedal triangle will be a Simson Line because the vertices
are on the circumcircle.

Finally if the pedal point P moves along the
circumcircle and we trace the Simson Line, we will get the following
deltoid figure. (Click **here**
for an animation of the following figure.)

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