Assignment 9

Pedal Triangles

By Carly Coffman

Let triangle ABC be any triangle. Then, if P is any point in the plane, then the triangle formed by constructing perpendiculars to the sides of ABC (extended if
necessary) locate three points R, S, and T that are the intersections. Triangle RST is the
Pedal Triangle for Pedal Point P.

Observations:

• When the pedal point, P, is located outside triangle ABC, the pedal triangle vertices become colinear when two of the pedal triangle vertices are at two of triangle ABC's vertices.

Pedal Point as the Centroid

First, we will explore what happens to the pedal triangle when the pedal point, P, is the centroid. The centroid is the intersection of the midpoints of the sides of a triangle.

Observations:

• When triangle ABC is isosceles, the pedal triangle is isosceles.
• The pedal point, P, does not go outside of triangle ABC.
• When the vertices of the pedal triangle are at the midpoints of the sides of triangle ABC, the pedal triangle is equilateral.
• When a vertex of the pedal triangle is at the vertex of triangle ABC, the pedal triangle only exists when the vertex angle of triangle ABC angle is a right or obtuse angle.

Pedal Point as the Incenter

Secondly, we will explore what happens to the pedal triangle when the pedal point is the incenter. The incenter is the intersection of the angle bisectors of a triangle.

Observations:

• When triangle ABC is isosceles, the pedal triangle is isosceles.
• When triangle ABC is a right isosceles triangle, the pedal triangle is isosceles.
• When triangle ABC is equilateral, the pedal triangle is equilateral.
• The vertices of the pedal triangle will not reach the vertices of triangle ABC until the degenerate case (triangle ABC becomes a line).
• The pedal point does not go outside of the triangle.

Pedal Point as the Orthocenter

Thirdly, we will explore what happens to the pedal triangle when the pedal point is the orthocenter. The orthocenter is the intersection of the triangle altitudes at the vertices.

Observations:

Notice that the pedal triangle is the orthic triangle when the pedal point is the orthocenter. Let's look at the definitions of the triangles to see why this is true.

Definitions:

• Orthic Triangle - triangle formed by the intersections of the altitudes of a triangle from the vertices
• Pedal Triangle - triangle formed by the intersections of the lines perpendicular to the sides and the pedal point

Since the pedal point is the orthocenter, the altitudes of the triangle at the vertices are on the line that is perpendicular to the sides through the orthocenter, or pedal point. Therefore, when the pedal point is the orthocenter, the pedal triangle is the orthic triangle.

• When triangle ABC is isosceles, the pedal triangle is isosceles.
• When triangle ABC is a right triangle, the pedal triangle vertices become colinear.
• When triangle ABC is obtuse, the pedal point exists outside triangle ABC.
• When triangle ABC is obtuse, the vetices of the pedal triangle are colinear.
• When triangle ABC is equilateral, the pedal triangle is equilateral.
• The only instance where a vertice of the pedal triangle is at a vertex of triangle ABC is when triangle ABC is a right triangle.

Pedal Point as the Circumcenter

Fourthly, we will explore what happens to the pedal triangle when the pedal point is the circumcenter. The circumcenter is the intersection of the perpendicular bisectors of the sides of a triangle.

Observations:

• When the pedal point is at the circumcenter, the vertices of the pedal triangle are the intersections of the perpindicular bisectors of the triangle sides.
• The pedal triangle is similar to triangle ABC since the vertices of the pedal triangle are the midpoints of the sides of triangle ABC. Angle BAC is congruent to angle TSR, angle ACB is congruent to angle TRS and angle CBA is congruent to angle STR.
• The pedal point, P, exists outside triangle ABC when triangle ABC is obtuse.
• The pedal point, P, exists on a side of triangle ABC when triangle ABC is a right triangle.
• The pedal point, P, exists inside triangle ABC when triangle ABC is acute.

Pedal Point on the Sides

Observations:

• Notice that the pedal point, P, becomes the pedal triangle vertex, R. If you were to place the pedal point on the side AC, P would be the same point as T. If you were to place the pedal point on the side BC, P would be same point as S.
• As the pedal point moves along the side of triangle ABC, the angle TPS stays the same measure.
• When P is placed at the vertex A or at the vertex B, the pedal triangle becomes degenerate (triangle becomes a line).
• When triangle ABC is a right triangle, the pedal triangle is a right triangle also.

Exploration

Locate the midpoints of the sides of the Pedal Triangle. Construct a circle with center at the circumcenter of triangle ABC such that the radius is larger than the radius of the circumcircle. Trace the locus of the midpoints of the sides of the Pedal Triangle as the Pedal Point P is animated around the circle you have constructed. What are the three paths?

GSP Solution

We get three elliptical loci from the midpoints of the pedal triangle.