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
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.
Orthocenter Inside the Triangle
Orthocenter Outside the Triangle
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 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.
When P lies on the circumcircle of ▲ABC, the Simson Line is formed.