Assignment 9

By: Olamide Alli

Exploration 1: How to Construct a Pedal Triangle
Exploration 2: Finding the Orthocenter, Circumcenter, and the Incenter of a Pedal Triangle.

Exploration 1
Step 1: Using GSP (Geometers Sketch Pad), construct a triangle ABC. Select a point (any point in the plane) and label it P. This point P will be your pedal point.

Step 2: Construct perpendicular lines from P to the lines formed. In order to construct this, extend each side of the triangle ABC (designated by blue dashed lines). The intersections of the perpendicular lines and the extensions of the sides of triangle ABC can be labeled as L, M, and N.

Step 3: The Intersection points, L, M, and N can be used as the vertices for the pedal triangle.

Step 4: To obtain a clearer depiction of the pedal triangle, hide the perpendicular lines initially constructed.

(Please note that the pedal point, P, can be moved as well in order to further any explorations and give a visual representation of what was just discussed).

Exploration 2
What is P, the pedal point, was the orthocenter of the triangle ABC? The pedal triangle (designated as LMN) is the orthic triangle for the triangle ABC.

The orthocenter for the triangle ABC can be moved to the outside of the triangle. Therefore the pedal triangle, LMN, makes an orthic triangle for the triangle ACP.

What if P, the pedal point, was the circumcenter of the triangle ABC? Now the pedal triangle, LMN, is the medial triangle for the triangle ABC. Notice that the vertices of the pedal triangle LMN are the midpoints of the sides of the triangle ABC.

If you move the circumcenter outside of the triangle ABC, the pedal triangle LMN will still be the medial triangle for the triangle ABC.

What if the pedal point, P, was the incenter of the triangle ABC?

When P is the incenter of the triangle ABC, then P is also the circumcenter of the triangle LMN. Remember that the circumcenter is also the intersection of the perpendicular bisectors of each side.