Pretty Pedal Triangles

Assignment 9

By Amber Candela

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The Problem

Explore and discuss pedal triangles.

What is a Pedal Triangle?

A pedal triangle is the triangle created by constructing perpendiculars to the sides of the triangle to any point P in the plane. There are different types of pedal triangles. Point P can be inside the triangle or outside the triangle.

When point P is inside the triangle

When point P is outside the triangle

If you would like to create your own pedal triangle, use the script tool below. In order to create the pedal triangle, first create triangle ABC, then place your point P where you would like it.

Pedal Triangle Script Tool

Special Cases

I explored four special cases of the pedal triangle.

• What if the pedal point was the orthocenter.
• What if the pedal point was the incenter.
• What if the pedal point was the circumcenter.
• What if the pedal point was the centroid.

Why these four points?

These four points are common centers of the triangle.

## Orthocenter as the Pedal Point

What if the pedal point is the orthocenter? Click below to see the script tool of the pedal triangle with the pedal point P being the orthocenter.

Pedal Triangle with P = Orthocenter

Orthocenter Inside the Triangle

Orthocenter Outside the Triangle

Orthocenter in a Right Triangle

The orthocenter of a triangle is the point of intersection of the altitudes of the triangle. Altitudes are the when the vertices are connected to the opposite sides by a perpendicular. When the orthocenter is inside the triangle, the pedal point will be collinear with the vertex of the original triangle and the vertex of the pedal triangle. This is because the altitude is perpendicular to the side, so where the pedal point is perpendicular to the side will be in the same line. When the pedal point is outside the triangle, the points are once again collinear because they are all perpendicular to the side of the triangle.

• ## Incenter as the Pedal Point

What if the pedal point is the incenter? Click below to see the script tool of the pedal triangle with the pedal point P being the incenter.

Pedal Triangle with P = Incenter

Incenter as the Pedal Point

Incenter in an Equilateral Triangle

The incenter of the triangle is the point of intersection of the angle bisectors. The incenter of the triangle is always inside the triangle which then makes the pedal triangle always inside the original triangle. If the triangle is an equilateral triangle, the pedal triangle is the medial triangle. This means that the pedal triangle is 1/4 the area of the original triangle.

• ## Circumcenter as the Pedal Point

What if the pedal point is the circumcenter? Click below to see the script tool of the pedal triangle with the pedal point P being the circumcenter.

Pedal Triangle with P = Circumcenter

Circumcenter Inside the Triangle

Circumcenter Outside the Triangle

The circumcenter of a triangle is the intersection of the perpendicular bisectors of the triangle.ince the circumcenter is already the perpendicular bisectors, using the circumcenter as the pedal point will create a pedal triangle whose vertices are the midpoints of the triangle. The pedal triangle is then the medial triangle of the original triangle. The medial triangle is the triangle constructed using the midpoints of the triangle. The pedal triangle will always remain inside the triangle even if the pedal point is outside of it as well as the vertices of the pedal triangle always being the midpoints of each side of the triangle. The pedal triangle created is similar to the original triangle. The sides are in the ratio of 2:1 and the angle measure is the same.

• ## Centroid as the Pedal Point

What if the pedal point is the centroid? Click below to see the script tool of the pedal triangle with the pedal point P being the centroid.

Pedal Triangle with P = Centroid

Centroid Image 1

Centroid Image 2

The centroid of a triangle is the intersection of the medians of the triangle. A triangle median is the line that joins the vertex to the midpoint of the opposite side. The centroid is always inside the triangle, but the pedal triangle will not always remain inside the triangle. One of the pedal points will go outside of the triangle. Can you figure out when that is?

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