Forum Geometricorum, 9 (2009) 27--39.

Abstract: 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 M_i on pairs of points (U,X). Many triangles homothetic to ABC are examined, and properties of the four mappings are presented. In particular, M_i(U,X) = M_i(X,U) for i = 1,2,3,4, and M_1(U, M_1(U,X)) = X; for this reason, M_1(U,X) is given the name U-vertex conjugate of X. In the introduction of this work,

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