### Assignment 9

# Pedal Triangles

#### By

## Jen Curro

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 locate three points R, S, and
T that are the intersections. Triangle RST is the PEDAL TRIANGLE
for PEDAL POINT P.

Now we will have a script of that here.

Now we are going to examine what happens
if our pedal point is in particular spots.

If the pedal point is the centroid of
the triange we get:

If the pedal point is the incenter we
get:

If we make P the orthocenter we get:

Even if the orthocenter is outside-(NOTE:
the pedal triangle is the orthic triangle when it is inside)

When P is the circumcenter of the triangle:

Even when the circumcenter is outside
the triangle as so:

When P is the center of the nine point
circle:

When P is on a side of the triangle:

When P is one of the vertices of the triangle:

There are some cases in which three vertices
of the Pedal triangle are colinear. This segment is then called
the Simson Line.

Locating the midpoints of the sides of
the Pedal Triangle and constructing a circle with center at the
circumcenter of triangle ABC such that the radius is larger than
the radius of the circumcirle and tracing the midpoints of the
sides of the Pedal triangle as Point P is animated around the
circle gives you three paths. Click
here

Repeating this with the path being on
the circumcirle also produces three paths. Click
here

When the lines of the Pedal Triangle are
traced, rather than the midponts we can then experiment with different
paths and those figures produced. In particular we want to examine
the circumcircle. Click here.

Now we are going to do the same process
only now our outside circle is less than our circumcircle and
look how it has changed, click here.

Other examinations can be made...what
can you come up with?

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