Assignment #9

Pedal Triangles

By

Michelle Nichols

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.

To construct the Pedal Triangle, draw triangle ABC and point P is any point in the plane (See picture above). Then construct the perpendiculars to sides AB, AC, and BC from the point P. The intersections of the perpendiculars and the sides are points R, S, T.  Our Pedal Triangle is the orange triangle, RST.

What if the Pedal Point P is the centroid of the triangle?

What if P in the incenter?

What if P is the Orthocenter?  Even if outside of ABC?

Orthocenter Inside the Triangle

Orthocenter Outside the Triangle

What if . . . P is the Circumcenter . . . ? Even if outside ABC?

Circumcenter Inside the Triangle

Pedal Triangle RST lies inside ▲ABC when point P is the circumcenter.

Circumcenter Outside the Triangle

If the Circumcenter is outside the Triangle, the Pedal Triangle is still inside ▲ABC

What if . . . P is the Center of the nine point circle for triangle ABC?

What if P is on a side of the triangle?

If P is on a side of the triangle, then the point P is shared with one of the vertices of ▲RST.

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

When P is one of the vertices of the triangle, a straight line is formed.  In the above sketch, A,P,R,S all share the same point.

Find all conditions in which the three vertices of the Pedal triangle are colinear (that is, it is a degenerate triangle). This line segment is called the Simson Line.

When P lies on the circumcircle of ▲ABC, the Simson Line is formed.