Assingment #9 is about pedal triangles. A __pedal triangle__ is a triangle formed by taking
a geven triangle ABC and a an arbitrary point in the plane P and
constructing the perpendiculars to the side of the triangle ABC
from point P. The resulting intersections form the vertices of
the pedal triangle. These vertices are connected and the pedal
triangle to triangle ABC is triangle RST is formed. Point P is
called the pedal point.

Explore Pedal Triangles on GSP by going here.

Now we will explore pedal triangles for the following cases:

1. Pedal point (P or G) is the centroid of the triangle. P is Centroid.gsp

2. P (or I) is the incenter of the triangle. Use tool table to construct new triangle.

3. P is the orthocenter of the triangle. P is Orthocenter.gsp

4. P is the circumcenter of the triangle. P is Circumcenter.gsp

5. P is the center of a nine point circle. P is Center Nine Pt Circle.gsp

6. P is on the side of the triangle. P is on Side of Triangle.gsp

7. P is on the vertices of the triangle. P is a Vertex of Triangle.gsp

Please click on the links to explore with GSP.

There is a special case involving pedal triangles
where the three vertices of the triangle are colinear. The pedal
triangle is a degenerate triangle. This is called the __Simson
Line__.

Explore the Simson Line with the following GSP script. This will allow you to see the envelope of the Simson Line.

How is the Simson Line and the segment connecting P to the orthocenter related? What is the relation of intersection?

Simson Line and Orthocenter.gsp

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