Department of Mathematics
J. Wilson, EMAT 6680

Assignment # 9
by Jan White

1a. 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 (extended if necessary) locate three points R, S, and T that are the intersections. Triangle RST is the Pedal Triangle for Pedal Point P.

1b. Use GSP to create a script for the general construction of a pedal triangle to triangle ABC where P is any point in the plane of ABC.
    Here is a script for a pedal triangle. It should end up looking like this.
 
 

2. What if pedal point P is the centroid of triangle ABC?  The vertices of triangle RST will end up on the segments of the triangle ABC.  For an animated sketch click here.

3. What if . . . I is the incenter . . . ?
4. What if . . . H is the Orthocenter . . . ? Even if outside ABC?
5. What if . . . C is the Circumcenter . . . ? Even if outside ABC?
6. What if . . . P is the Center of the nine point circle for triangle ABC?
7. What if P is on a side of the triangle?
8. What if P is one of the vertices of triangle ABC?

For questions 3-7 the vertices of the pedal triangle RST will all end up on some side of the triangle ABC as long as the orthocenter and circumcenter are in ABC. If the orthocenter and circumcenter are outside ABC then the pedal triangle RST will also be outside.  If the pedal point moves to one of the vertices then RST degenrates to a line segment.

For visual answers to these questions click on this animation.

9. Find all conditions in which the three vertices of the Pedal triangle are colinear (that is, it is a degenerate triangle). This line segment is called the Simson Line.

As we can see the Simpson Line can be observed as the pedal point moves around the cicumcircle of triangle ABC.  For an animation click here.