Pedal Triangles
Jamila K. Eagles
Let 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 and locating the
points of intersection, R, S, and T, Triangle RST is said to be a pedal
triangle with pedal point P. (The construction is in green in the figure
below).
We can see the different pedal triangles formed by the centers of triangle
ABC in the figure below.
Red= Circumcenter(C), Green= Centroid(G), Light Blue=Nine Point Circle
Center(N), Dark Blue = Incenter(I), Pink = Orthocenter(H).
Click Here to see the exploration of the location of the pedal triangles
as the size of triangle ABC varies.
The case in which the pedal triangle is a degenerate triangle, meaning
that the vertices of the pedal triangle are colinear, is called the Simson
Line.
Through my exploration I found that the pedal triangle is the simson
line when the pedal point P lies any vertex of the triangle, and on the
circumcircle of the triangle. It follows logically since the three vertices
of a triangle lie on the circumcircle.
Tracing the lines that form triangle RST, where P is located at vertex
A as the pedal point P travels around the circumcircle, we get the figure
below.
The picture shows that the locus of the lines that for the simpson line
is a triangle.