Mappings Associated with Vertex Triangles

Methods of linear algebra are applied to triangle geometry. The vertex triangle of distinct circumcevian triangles is proved to be perspective to the reference triangle ABC, and similar results hold for three other classes of vertex triangles. Homogeneous coordinates of the perspectors define four mappings Mi on pairs of points (U, X). Many triangles homothetic to ABC are examined, and properties of the four mappings are presented. In particular, Mi(U,X) = Mi(X, U) for i = 1, 2, 3, 4, and M1(U,M1(U, X)) = X; for this reason, M1(U, X) is given the name U -vertex conjugate of X. In the introduction of this work, point is defined algebraically as a homogeneous function of three variables. Subsequent definitions and methods include symbolic substitutions, which are strictly algebraic rather than geometric, but which have far-reaching geometric implications.