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.



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