nL-n-Tf5   nL-Linear Polar

nL-n-Tf5 is the generalization of the trilinear polar (also called tripolar) in a triangle.
nL-n-Tf5 transforms a random point P into an “nL-Linear Polar”, which is a line in an n-Line.
In a 2-Line it is the line that forms a harmonic bundle with the lines of the 2-Line and the line connecting P and the only intersection point of the 2-Line.
In a 3-Line it is the well-known Trilinear Polar in the Triangle, also called Tripolar. See Ref-13.
In a 4-Line it is the OL-Trilinear Polar (QL-Tf6) in the Quadrilateral which was found by Tsihong Lau. See Ref-34, QFG-messages #2154 and #2161. It is derived from the Tripolar.
The method for constructing the nL-Linear Polar in an n-Line for n>4 is done in a recursive way by using the same “upgrading-method” as used for deriving the n=3 case. See figure below.
nL n Tf5  nL Linear Polar 02
Correspondence with ETC/EQF:
When n=3, then nL-n-Tf5 = Trilinear Polar, also called Tripolar
When n=4, then nL-n-Tf5 = QL-Tf6.


nL-n-Tf5 is the dual of nP-n-Tf2, because when the notions of “Points” and “Lines” are interchanged in the description of the construction and “the intersection of two lines” and “the connection of two points” are interchanged, then the other transformation pops up, which makes them duals.