Assignment #9

by Carol Sikes

Properties of Pedal Triangles

Introduction

In this write-up, I explore the relationship between pedal triangles and to a given triangle based on the location of the pedal point.

Consider a triangle ABC and a point P in the plane. This point P is known as the pedal point, and the pedal triangle is constructed from the intersections of the perpendiculars through P to the sides of triangle ABC. To use a GSP script to create a Pedal triangle given a triangle ABC and a point P, click here. In the figure shown below, triangle RST is the Pedal triangle of ABC for point P.

The location of the Pedal triangle changes as the point P is moved around in the plane. When P lies outside the circumcircle of ABC, at least some portion of triangle RST is external to triangle ABC. When P lies on the circumcircle of ABC, points R, S, and T become colinear and RST is a degenerate triangle. If the point P lies inside the circumcircle, at least some portion of triangle RST is internal to ABC. When P lies on the sides of triangle ABC or inside ABC and ABC is an acute triangle, triangle RST is internal to ABC. When ABC is an obtuse triangle, this is not always true. To experiment using GSP to demonstrate the effects of moving the point P to different points in the plane, including some of the centers of triangle ABC, click here. The Pedal triangle when P is the center of the nine-point circle is shown below.

Conjectures About the Loci of Midpoints of the Pedal Triangle

The loci of the midpoints of the sides of Pedal triangles formed when P lies on a circle that is concentric to the circumcircle are ellipses. This is true even when P lies on the circumcircle, and the Pedal triangle becomes the Simson line. Based on my observations in GSP, I have made several conjectures concerning these ellipses:

It appears that the ellipses formed as the loci of the midpoints of the Pedal triangle RST have major axes that are parallel to the angle bisectors of triangle ABC when ABC is an acute triangle.
When ABC is an obtuse triangle, two of the major axes appear parallel to the angle bisectors, and the third ellipse has a minor axis that appears to be parallel to the angle bisector of the obtuse angle.
It also appears that the width of the elliptical axes that are perpendicular to the angle bisectors of triangle ABC are proportional to the measures of the angles of triangle ABC. For example, when ABC is an acute triangle with angles of similar measure, the width of the minor axes of the three ellipses formed as loci of the midpoints of the Pedal Triangle is very similar. When ABC is an obtuse triangle, the ellipse formed with an axis parallel to the angle bisector of the obtuse angle has a much wider axis perpendicular to the angle bisector than the other two ellipses.
When triangle ABC is a right triangle, the locus of midpoints formed around the vertex of the right angle is a circle with its center at the midpoint of the segment between this vertex and the circumcenter.

To observe the loci of midpoints of the Pedal Triangle formed when P is moved about one of the three circle shown below, click here.

Conjectures About the Simson Line

As mentioned above, when the Pedal point lies on the circumcircle of triangle ABC, the Pedal triangle is a degenerate triangle. Points R, S, and T are collinear and they form what is known as the Simson line. There are some interesting properties related to the Simson line.

The first conjecture is related to the Pedal points on the circumcircle that produce Simson Lines that are concurrent with one of the sides of the triangle ABC.

General Conjecture: When P lies at the intersection of the circumcircle and the line through a given vertex and the circumcenter, the Simson Line is concurrent with the side opposite the given vertex.

For example, refer to the figure below:

When P lies at the intersection of the circumcircle and the line through A and the circumcenter (point X), the Simson Line is concurrent with side BC.
When P lies at the intersection of the circumcircle and the line through B and the circumcenter (point Y), the Simson Line is concurrent with side AC.
When P lies at the intersection of the circumcircle and the line through C and the circumcenter (point Z), the Simson Line is concurrent with side AB.

Next we consider a Pedal Triangle with P located on the circumcircle and the line segment constructed between P and the Orthocenter of ABC, point H.

General Conjecture: As P moves about the circumcircle, the Simson line bisects the segment PH.

And finally we look at the intersection of two Simson lines created from separate Pedal points located on the circumcircle of triangle ABC. Let P, P', and X be three distinct points on the circumcircle as shown below.

General conjecture: The angles formed at the intersection of the two Simson lines formed from Pedal points P and P' are equal to the angular measures of the arcs formed between P and P'.
For example, the acute angle of intersection of the two Simson lines below is equal to the measure of angle P'XP.