 Pedal Triangles Exploration

By Mary Negley

For this assignment I will explore pedal triangles and demonstrate what happens when the pedal point is animated around different paths.  First things first!  What is a pedal triangle?

A pedal triangle is created by constructing lines from a point P, which is anywhere in the plane, perpendicular to the sides of a triangle ABC.  The intersections of the lines with the sides of the original triangle are the vertices of the pedal triangle.  Click here for a GSP sketch, where you will be able to move around point P and see many possible pedal triangles. The first problem, with which I am faced, is the following:

Animate the pedal point P about the incircle of ABC.  Trace the loci of the midpoints of the sides.  What curves result?  Repeat if ABC is a right triangle.

To find the answer, I must first know what an incircle is.  An incircle is, by definition, the circle that is tangent to all three sides of a triangle and its center is the incenter of the triangle, which is the concurrence point of the angle bisectors.  Below is a sketch of the triangle with its incircle. Now I will animate P about the incircle and trace the midpoints of the sides of the pedal triangle.  Click here for GSP sketch with animation. It appears that ellipses result.

What happens if Triangle ABC is a right triangle?

Click here for GSP sketch with animation.

The following result occurs: The traces still create two ellipses, but a circle is also created.

My next problem is as follows:

Construct an excircle of triangle ABC.  Animate the pedal point P about the excircle and trace the loci of the midpoints of the sides of the pedal triangle.  What curves result?  Look at the angle bisectors through the center of the excircle.  How are the loci positioned with respect to the angle bisectors?

This time I need to know what an excircle is.  An excircle is, by definition, centered at the intersection of the external angle bisectors of two vertices and tangent to the side between those two vertices and also tangent to the extension of the other two sides.  An example is given below. Now I will pick a random point on the excircle to be the P and I will create a pedal triangle based on that point. Like in the last problem, I will trace the midpoints of the sides as P goes along the circle.  Click here for a GSP sketch with animation. Like last time it appeared to make ellipses.  To answer the second part of the problem (Look at the angle bisectors through the center of the excircle.  How are the loci positioned with respect to the angle bisectors?), I need to show the angle bisectors again. Notice that the intersection of the exterior angle bisectors is surrounded by two of the ellipses and the other ellipse contains the part of an exterior angle bisector.