There is an aspect of the trilinear polar tr(K) of a point K with respect to a triangle ABC connecting it with homographies. In fact, the system (ABC,K) of a triangle and a point uniquely defines a homography F (perspectivity) fixing K and mapping {A,B,C} to the traces {A',B',C'} of K.
This homography is conjugate F=G*F0*G-1 to the antihomothety F0 of the equilateral to its medial triangle. G is the projectivity mapping the system (A0B0C0,O) of the equilateral triangle and its center to (ABC, K). It follows immediately that F is a homology (see Perspectivity.html ) with coefficient k=-2 having one isolated fixed point K and a line consisting entirely of fixed points which is no other than the trilinear polar tr(K) with respect to ABC.
The figure shows part of the sequence of triangles resulting by applying F and F-1 repeatedly to ABC. All resulting triangles have their vertices on the lines {KA,KB,KC}. The sides of all these triangles pass through three points {A0,B0,C0} on the trilinear polar tr(K).
Applying repeatedly F and F-1 to the circumconic c0 of ABC with perspector K (see TriangleConics.html ) we obtain also a sequence of conics in which is also contained the dual c1 of c0. These two sequences and their properties are discussed in some extend in IsotomicGeneral.html .
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Trilinear polar and homographies ![[0_0]](TrilinearProjectivity_files/TrilinearProjectivity_0_0_0.png)
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