Assignment 9: Pedal triangles

Ebru Ersari

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.

When Pedal Point is outside of the triangle ABC, the Pedal triangle looks like the triangle RST below.

When Pedal Point is inside of the triangle ABC, the Pedal triangle looks like the triangle RST below.

What if P is the centroid? Here is the pedal triangle when P is the centroid.

When P is the centroid, the pedal triangle is always inside of the triangle ABC.
What if P is the incenter? Here is the pedal triangle when P is the incenter.

When P is the incenter, the pedal triangle is always inside of the triangle ABC.
What if P is the Orthocenter and inside of the triangle ABC? Here is the pedal triangle when P is Orthocenter (P is inside).

When P is the Orthocenter and inside of the triangle ABC, the pedal triangle is inside of the triangle ABC, as well.
What if P is the Orthocenter and outside of the triangle ABC? Here is the pedal triangle when P is Orthocenter (P is outside).

When P is the Orthocenter and outside of the triangle ABC, part of the pedal triangle is outside of the triangle ABC, too.
What if P is the Circumcenter? Here is the pedal triangle when P is the Circumcenter (P is inside).

When P is the Circumcenter and inside of the triangle ABC, the pedal triangle is inside of the triangle.
What if P is the Circumcenter? Here is the pedal triangle when P is the Circumcenter (P is outside).

When P is the Circumcenter and outside of the triangle ABC, the pedal triangle is inside of the triangle. Thus, if P is the Circumcenter, pedal triangle is always inside of the triangle regardless of the pedal pointâ€™s location.
What if P is the center of the nine point circle for triangle ABC?

When P is inside of the triangle ABC, the center of the nine point circle is inside of the triangle. In this case, the pedal triangle is also inside of the triangle ABC.
What if P is on a side of the triangle?

When P is on a side of the triangle ABC, P becomes one of the vertices of the pedal triangle. The pedal triangle is inside of the triangle ABC.

What if P is one of the vertices of triangle ABC?

When pedal point is one of the vertices of the triangle ABC, the pedal point becomes one of the vertices of the triangle ABC. There is no pedal triangle in this case.