#### EMAT 6680

#### Cathryn Brooks

#### Assignment 9

#### Pedal Triangles

#### Pedal Triangles

#### Definition: Let Triangle ABC be any triangle. Then if P is
any point in the plane. Constuct perpendiculars to the sides of
ABC (extended if necassary). Let R,S, T be the three points where
the the perpendiculars and the sides (or extended sides) of ABC
meet. RST is the Pedal Triangle defined
by P, the Pedal
Point, and triangle ABC.

Consider a triangle ABC. Construct its Incircle. Construct
a point **P** on the incircle. Construct the Pedal triangle
associated with point P.

We can now find the locus of the midpoints of the sides of
the Pedal Triangle (the aqua triangle).

The locus of the midpoints of the pedal triangle appear to
be ellipses with the foci on the lines used to find the center
of the incircle. These lines are the angle bisectors of the angles
of the triangle.

How do we expect the loci to change as the angles change?

First let's consider the angle BAC.We will start with a small
angle and increase it to see what happens to the locus on that
angle bisector (the purple curve).

A

As we see the purple locus is very narrow, almost a line. Increasing
angle BAC:

A

The locus of points is getting "fatter".

A

The locus is "fatter" still.

A

The locus appears to be a circle now. This would suggest the
foci get closer to each other on the angle bisector until they
meet when the angle being bisected is 90 degrees. Many relationships
might be explored to discover the underlying principles of this
problem.

To Experiment with the problem

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