Sarah Donaldson

EMAT 6680

12/4/06

Assignment 9 Write-Up

Problems 1-6: Pedal Triangles

In this write-up, I shall bring together some of the ideas about Pedal Triangles that I have learned in investigating problems 1-6 in this assignment. These problems require explorations of what happens with different placements of the Pedal Point.

I began in each case with ∆ABC and constructed its Pedal Triangle. From a Pedal Point, P, I drew perpendiculars to the sides (or sides extended) of ∆ABC. Each of these three intersections (D, E, and F) is a vertex of the Pedal Triangle.

From this basic construction, I proceeded to investigate the following questions posed in the assignment: What if P is the centroidof ∆ABC? the incenter of ∆ABC? the orthocenter of ∆ABC? the circumcenter of ∆ABC? the center of the nine point circle for ∆ABC?

After investigating each of these (see the accompanying GSP sketch), I found that the most interesting results occurred when the Pedal Point was placed either at the orthocenter or the circumcenter of ∆ABC.

When the Pedal Point is the same as the orthocenter of ∆ABC, the Pedal Triangle is same as the orthic triangle. This always happens because the orthic triangle and the Pedal Triangle are both created by feet of altitudes of ∆ABC. In the sketch below (as well as in the GSP sketch, page “4 orthocenter”), the Pedal Triangle is shown in red, and the orthic triangle is in green.

When the Pedal Point is the same as the circumcenter of ∆ABC, there are some special things about the Pedal Triangle. In summary, the Pedal Triangle is the same as the medial triangle. Here’s why:

The lines from the Pedal Point to the sides of ∆ABC are perpendicular to these sides (by definition of the Pedal Point). Additionally, the lines from the circumcenter to the sides of ∆ABC are the perpendicular bisectors of these sides (by definition of circumcenter). Since in this case the Pedal Point is the same as the circumcenter, we know that the vertices of the pedal triangle (D, E, and F) are the midpoints of the sides of ∆ABC. Therefore the triangle formed by D, E, and F is not only the pedal triangle of ∆ABC, but its medial triangle as well!

This means that ∆DEF is similar to ∆ABC, since a triangle is always similar to its medial triangle (see proof in GSP sketch page “5 circumcenter”).