Pedal Triangles

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 1^{st} pedal triangle are the
feet of the altitudes of the original triangle, thus the 1^{st} pedal
triangle is also the orthic triangle of the original.

Now letŐs move P to the circumcenter! Click here!

The vertices of the 1^{st} pedal triangle happen to
be the midpoints of segments AB, BC, and CA, thus the 1^{st} 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 1^{st} pedal
triangle we can see that P has become the circumcenter of the 1^{st}
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 3^{rd} pedal triangles we see that the
corresponding sides have the same ratio thus, our original triangle is similar
to the 3^{rd} pedal triangle.