Pedal Triangle Exploration

by

Chad Crumley

 

This is an investigation of merging the pedal point to different centers of a given triangle. 

 

First, here is triangle ABC with its pedal triangle XYZ and pedal point P.

 

  1. What happens when the pedal point merges with the centroid of triangle ABC? 

 

 

2.  What happens when the pedal point merges with the incenter of triangle ABC?

 

 

  1. What happens when the pedal point merges with the orthocenter of triangle ABC (inside)?

 

 

  1. What happens when the pedal point merges with the orthocenter of triangle ABC (outside)?

 

 

Conclusion:  It appears that for #1, 2, and 3 the vertices of the pedal triangle will lie on the given triangle ABC.  For #3 and 4, it appears that the vertices of the pedal triangle will lie on the altitudes of the given triangle ABC, regardless of the orthocenter being inside or outside triangle ABC.

 

What happens when the pedal point lies on a vertex of triangle ABC?  Click here to explore.

 

What happens if you trace the vertices of the pedal triangle while moving a vertex of the original triangle?  Click here to explore.  The trace leaves a circle and a line through the circle.  Two vertices of the pedal triangle are always on the circle.  Did you get something like this?

 


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