While the pedal point, Cevian point, and even pedal-Cevian point are commonly used concepts in triangle geometry, there seems to be no established term to describe the partitioning of an original triangle into three subtriangles , , and by the selection of a point . In this work, this process will be called triangulation (by analogy with more general use of that term), and the point used to construct such a triangulation will be called a triangulation point.
There is a remarkable series of theorems involving the triangles produced from an original triangle by triangulation. Let be a triangle center that is also a triangulation point, and call three triangles produced from an initial triangle by this point the triangulation triangles of . Then a remarkable number of triangle centers obey the following theorem: If is a triangle center of a triangle and , , and are the corresponding centers of the triangulation triangles of with respect to , then the lines , , and concur. A number of special cases are summarized in the table below (Hatzipolakis 1999).
is the isogonal conjugate of the complement of the symmedian point .