Triangulation Point

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TriangulationPoint

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 DeltaABC into three subtriangles DeltaABP, DeltaBCP, and DeltaCAP by the selection of a point P. In this work, this process will be called triangulation (by analogy with more general use of that term), and the point P 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 P be a triangle center that is also a triangulation point, and call three triangles produced from an initial triangle DeltaABC by this point the triangulation triangles of P. Then a remarkable number of triangle centers obey the following theorem: If P is a triangle center of a triangle DeltaABC and P_A, P_B, and P_C are the corresponding centers of the triangulation triangles of DeltaABC with respect to P, then the lines AP_A, BP_B, and CP_C concur. A number of special cases are summarized in the table below (Hatzipolakis 1999).

P point of concurrence triangle center function

circumcenter O Kosnita point Ko sec(B-C)

first Fermat point X first Fermat point X csc(A+1/3pi)

incenter I Kimberling center X_(1127) sin(1/4A)csc(3/4A)

symmedian point K Kimberling center X_(251) a/(b^2+c^2)

triangle centroid G triangle centroid G 1/a

X_(251) is the isogonal conjugate of the complement of the symmedian point K.