By Colleen Garrett
In this investigation I will explore how pedal triangles change as P moves to the orthocenter, circumcenter and incenter of the original triangle.
LetŐs first look at what happens when P moves to the orthocenter.
The vertices of the 1st pedal triangle are the feet of the altitudes of the original triangle, thus the 1st pedal triangle is also the orthic triangle of the original.
Now letŐs move P to the circumcenter! Click here!
The vertices of the 1st pedal triangle happen to be the midpoints of segments AB, BC, and CA, thus the 1st pedal triangle is also the medial triangle.
Now letŐs look at what happens when P is the incenter. Click here!
If we construct the circumcircle of the 1st pedal triangle we can see that P has become the circumcenter of the 1st pedal triangle.
Can we show that the third pedal triangle is similar to the first?
Using GeometerŐs sketchpad to calculate the ratio of the sides of the original and 3rd pedal triangles we see that the corresponding sides have the same ratio thus, our original triangle is similar to the 3rd pedal triangle.