## Assignment 9

## The Midpoints of a Pedal Triangle

For this following exploration, we need to first construct
the pedal triangle for a pedal point and observe the behavior
of the midpoints of that triangle. To construct the pedal triangle,
following these steps:

1) Draw a triangle ABC (extend the three segments into lines)

2) Place a pedal point, P outside the triangle

3) Construct perpendiculars to each of the three lines: AB,
BC, AC

4) Let the intersections be points: R,S, and T

5) You now have triangle RST as the pedal triangle.

For the GSP sketch and script tool click
here ->

Now, for the exploration, a circle is drawn where the radius
is greater than the radius of the circumscribed circle of triangle
ABC. The midpoints of the pedal triangle are constructed.

Before we jump into any exlporation, it would be helpful to
think about the types of relationships we are looking for. We
can also divide our exploration into three parts with respect
to an acute, obtuse, or right parent triangle ABC.

What would the locus of points look like for each case, obtuse,
acute, and right? What kind of relationships can we develop with
our knowledge of pedal points, pedal triangles, etc....

Now, we are ready to explore the GSP sketches.

To explore each case click below:

Right Triangle case (with
java sketchpad ) OR (without
java sketchpad).
Obtuse Triangle case (with
java sketchpad ) OR (without
java sketchpad).
Acute Triangle case (with
java sketchpad ) OR (without
java sketchpad).

## Observations:

After viewing each different case, the following observations
could made and further explored:

For the Right triangle case, the path of the midpoints of the
pedal triangle seem to construct two ellipses and a circle. For
the obtuse triangle case, locus of points seem to construct three
ellipses. Similarily, for the acute triangle case, the locus of
points seem to construct three ellipses as well. But, for the
acute case, the ellipses are "thinner" than the obtuse
case. This would provoke the observer to correlate some kind of
relationship between the angle measure and eccentricity of the
locus of points.

**Obtuse case **versus **Acute case**:
Note that the red, blue, and yellow ellipses are the locuse
of points for the midpoint of the pedal triangle.

**Right Case**

Return to Assignment Matrix click
here ->

Questions? E-mail: **gt0353d@arches.uga.edu**