This is the writeup of Assignment #9 
Brian R. Lawler

EMAT 6680 
01/02/01

See the Definition and Construction of a Pedal Triangle  
II.

Exploring conditions of the Pedal Triangle 
Considering various loci and traces 
(Click on image to explore in Geometer's Sketchpad.)
2.  What if pedal point P is the centroid of triangle ABC?  
When the pedal point is located at the centroid of triangle ABC, it appears that the pedal triangle will always exist inside the original triangle.  
3.  What if . . . P is the incenter . . . ?  
When the pedal point is located at the incenter of triangle ABC, points R, S, and T lie at the points of tangency of triangle ABC's incircle. Thus, this incircle is also the circumcircle of triangle RST.  
4.  What if . . . P is the Orthocenter . . . ? Even if outside ABC?  
When the pedal point is located at the orthocenter of triangle ABC, the points R, S, T occur at the intersection of triangle ABC's sides and altitudes.  
5.  What if . . . P is the Circumcenter . . . ? Even if outside ABC?  
When the pedal point is located at the circumcenter of triangle ABC, the points R, S, T occur at the midpoints of triangle ABC's sides.  
6.  What if . . . P is the Center of the nine point circle for triangle ABC?  
Recall triangle ABC must be acute for the nine point center to exist. When the pedal point is located at this center, the points R, S, and T all lie on the sides of triangle ABC.  
7.  What if P is on a side of the triangle?  Click here to investigate the GSP file. 
Again, R, S, and T will always remain on triangle ABC.  
8.  What if P is one of the vertices of triangle ABC?  
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.  
The pedal triangle degenerates as P becomes one of the points A, B, or C. 
Comments? Questions? email me at blawler@coe.uga.edu 
Last revised: January 2, 2001 