In the following figure the basic facts for such a system (ABC,K) are discussed. The properties result from corresponding properties that are valid and trivial to show for equilateral triangles. The figure results by mapping the system (A

[1] ABC denotes the triangle of reference. A'B'C' the triangle of

[2] The trilinear polar of K with respect to any of these systems is the same line PP' (PP'=tr(K)=tr'(K)=tr''(K)).

[3] The tripoles tr(PK) of lines through K generate a circumconic c

[4] Line tr(K) is the polar of K with respect to any one of these conics.

[5] For every point P on tr(K) points {t(P),t'(P),t''(P)} are on the line passing through K and tr(PK).

[6] For every point P on tr(K) lines {tr(P),tr'(P)} are tangents correspondingly to {c

[7] For every point P on tr(K) lines tr(t(P)) and tr(t'(P)) coincide with a line through K, which is the common polar of P with respect to both conics {c

Sequence S can be extended also inside A'B'C' by taking successive traces with respect to K. The trilinear polar of K with respect to every triangle of this sequence is the same line. This extends [1.2] for the whole of the sequence S.

{t

This homography is conjugate F=G*F

The various triangles of the sequence S of section 2 are simply the result of repeated application of F or F

Analogously the sequence of conics is created by applying repeatedly F or F

See the file TrilinearProjectivity.html for a figure displaying part of the sequence of triangles and the sequence of conics.

IsotomicChart.html

IsotomicConicOfLine.html

Perspectivity.html

TrilinearProjectivity.html

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