Sarah Donaldson

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”).

Ideas for classroom use:

Exploring Pedal Triangles is a good extension of a series of lessons revolving around points of concurrency of triangles. This particular investigation would be a good one for either a high school geometry course (assuming there’s time) or a course for pre-service teachers. In either case, there would be great value in focusing in on any of the points of concurrency as they relate to Pedal Triangles.

For example, What happens when the Pedal Point is the same as the circumcenter of ∆ABC? Students should be guided first through construction of a Pedal Triangle, of course. In doing so they would become familiar with the nature of it: the vertices of the Pedal Triangle lie on the points of intersection of the sides (or extended sides) of ∆ABC and the lines perpendicular to these sides (or extended sides) through the Pedal Point, P. This construction lays a good foundation for understanding particular Pedal Triangles, as the one formed when the Pedal Point is the circumcenter of ∆ABC.

Students should already know that the circumcenter of ∆ABC is the point of concurrency of the perpendicular bisectors of the sides of the triangle. If this circumcenter is to be used as the Pedal Point, then points D, E, and F (the vertices of the Pedal Triangle) must be the same as the midpoints of the sides of ∆ABC (as is explained above). Since the medial triangle, by definition, has its vertices at the midpoints of the sides of ∆ABC, the Pedal Triangle just constructed must also be the medial triangle.

Beyond this (if it hasn’t been explored elsewhere in the course), students could explore features of the medial triangle (its similarity to ∆ABC, the ratio of its area to the area of ∆ABC, etc.).