Explorations with the Pedal Triangle

By

Doug Westmoreland

This write-up consists of several problems from assignment #9.

First, lets take a look a the definition of a pedal triangle. A pedal triangle is a triangle formed by taking a given triangle ABC and any point P in the plane and constructing the perpendiculars to the sides of triangle ABC (extend if necessary) from point P. These intersections on the sides of triangle ABC determine the vertices of the pedal triangle. Point P is called the pedal point. In the figure below, triangle RST (red) is the pedal triangle to triangle ABC (blue). Point P is the pedal point.

For a GSP sketch that you drag points A, B, or C around and observe the behavior of the pedal triangle, click here.

For a GSP script that will construct the pedal triangle, click here.

Now, lets take the pedal point of triangle ABC and place it in special locations. These locations will be the a) centroid, b) incenter, c) orthocenter, d) circumcenter, e) center of a nine point circle, f) side of the triangle, g) vertex of triangle ABC.

Click on any of the GSP sketches listed below to observe the behavior of the pedal triangle at the specified location:

How did the pedal triangle behave in the sketches above? What did you notice?

One should have discovered that the pedal triangle becomes a degenerate triangle when the pedal point is moved to any of the vertices of triangle ABC. Do you think that there are other points in the plane that the pedal point could be and the pedal triangle would be a degenerate triangle? Go back to any of the GSP sketches above and try dragging the pedal point around in the plane and find points (other than the vertices of triangle ABC) where the pedal triangle would be a degenerate triangle. What do you think that the locus of these points would be?

There is indeed other points in the plane that the pedal point could be and produce a degenerate pedal triangle. The locus of these points would be the circumcircle of triangle ABC. Click here for a GSP sketch in which you can drag the pedal point to various locations on the circumcircle and observe the pedal triangle.

In each of the drawings above the pedal lies on the circumcircle of triangle ABC. Notice the the pedal triangle becomes a degenerate triangle (red segment) in each case. Click here for GSP sketch in which the pedal point is animated around the circumcircle of triangle ABC. When the pedal point is animated around the circumcircle, you will see all conditions in which the three vertices of the Pedal triangle are collinear. This line segment is called the Simpson Line.

Click here for a GSP sketch of the envelope of the Simpson Line as the pedal point moves along the circumcircle.

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