Assignment 9: The Pedal Triangle

by

Melissa Silverman

To construct a pedal triangle, begin with a triangle (whose sides are lines, not line segments). Pick any point on the plane, Point P. Construct perpendicular lines from the sides of the triangle to Point P. Mark the intersections of those perpendicular lines and the sides of the triangle. Those three points of intersection connect to form the pedal triangle. In the construction below, the pedal triangle is drawn in red.

Using GSP to make a pedal triangle allows one to move Point P on the plane and observe how the position of Point P affects the pedal triangle. For example, what happens if Point P is merged with the centroid?

The pedal triangle becomes the medial triangle when Point P is merged with the centroid. The pedal (or medial) triangle divides the triangle in to 4 congruent triangles that are similar to Triangle ABC.

We can also investigate what happens when Point P merges with the orthocenter or circumcenter?

The pedal triangle is inside Triangle ABC, cutting it into 4 triangular parts. The parts are neither congruent to each other nor similiar to Triangle ABC.

When Point P is merged with the circumcenter, the pedal triangle is again inside Triangle ABC, cutting into 4 triangular parts. The parts are not congruent.

If Point P is on the circumcircle, the pedal triangle morphs into a straight line. Similarly, if Point P is on one of the vertices of Triangle ABC, the pedal triangle is a straight line because the vertices are a part of the circumcircle.

Above, notice that when Point P is on the circumcircle, the pedal triangle becomes a straight line. Similarly, when Point P is on a vertex, the pedal triangle is a straight line because the vertex is part of the circumcircle.

If you trace the pedal triangle line as Point P is moved along the circumcircle, the result is the envelope of the Simson line.