Pedal Triangles

by

Rob Walsh

Given a triangle and a point, a pedal triangle is constructed by using the intersections of the perpendiculars from the point to each side of the triangle as vertices. The given point is known as the pedal point.

What we are interested in are special cases of this point. But first, a look at a gsp pedal triangle construction

Regardless of where the pedal point is located inside or outside of the triangle, a pedal triangle can be constructed. However, as this already looks similar to several constructions we have doen in the past, let's consider special cases of the pedal point's location.

Pedal point as a triangle center

My first set of explorations involved constructing a pedal triangle where the pedal point was each of four centers. When I constructed a pedal point onto a triangle's centroid and incenter, nothing particularly interesting happened. However, when I moved the pedal point onto the triangle's orthocenter, the pedal triangle became the orthic triangle. Similarly, when I moved the pedal point onto the circumcenter, the pedal triangle became the medial triangle. You should explore for yourself.

Centroid

Incenter

Orthocenter

Circumcenter

Pedal point on a triangle

My second set of explorations involved constructing a pedal triangle where the pedal point was on the triangle. Read ahead for the results or explore for yourself.

Pedal point on a triangle's side

When I moved the pedal point onto the triangle's side, the point became one of the vertices of the pedal triangle. Exploration into this yielded the same results, no matter what side it was on or where on the side it was placed!

Pedal point on a triangle's vertex

When I slid the pedal point up one of the sides to a vertex of the triangle, the pedal triangle disappeared and what was left was one of the altitudes. Observe above, pedal point P of triangle ABC becomes vertex B and the altitude is formed that drops to foot AC. A similar thing happens when pedal point P is moved to vertex A and vertex B as well!

If you want to see what happens when you place the pedal point in several other places, explore for yourself!!!