Clark Kimberling, Mappings associated with vertex triangles,
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, 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.
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