Pedal Triangle
The pedal triangle of with respect to
The pedal triangle of a triangle and point is the triangle whose vertices are the projections of a to the sides of the triangle. For instance, when is the orthocenter of the triangle, the pedal triangle is the orthic triangle; when the term is used without further qualification, this is often what is meant by "the" pedal triangle of .
Contents
Properties
Any pedal triangle satisfies
If lies on the circumcircle of the triangle, the pedal triangle becomes the Simson line with respect to . This is useful both for showing that lies on the circumcircle, and for showing three points are collinear.
Further,
The area of the pedal triangle is given by
where is the radius of the circumcircle, and is the circumcenter.