QA-Tf7: QA-Trilinear Pole

 
Let P1.P2.P3.P4 be a quadrangle. Let L be a random line. Let T1 be trilinear pole of L wrt triangle P2.P3.P4.
Similarly, we have T2, T3, T4. Then lines P1.T1, P2.T2, P3.T3, P4.T4 are concurrent.
This transformation was found by Tran Quang Hung. See Ref-34, QFG#2084.
It is a transformation with very simple coordinates.
The equivalent of this transformation in a quadrilateral is QL-Tf6.
QA-Tf7 and QL-Tf6 have corresponding coordinates (interchange (p:q:r) with (l:m:n)).
 
QA Tf7 TripolarTransformation 02
 
Note that the Quadrangles P1.P2.P3.P4 and T1.T2.T3.T4 are perspective with perspector QA-Tf7(L) and perspectrix L. So this transformation maps the perspectrix into the perspector of this perspective quadrangles.
Given the perspector there is a reverse transformation mapping the perspector into 3 possible perspectrices. For details see Ref-34, QFG#2139. 
 
CT-coordinate:
Given Line L(x:y:z), then the coordinates of QA-Tf7(L) are:
(y z (2 p x + q y + r z) : x z (p x + 2 q y + r z) : x y (p x + q y + 2 r z))
 
Properties:
QA-Tf7(L) is the concurrent point of the 3 versions of QL-Tf2(L). See QA-3QG1.
QA-Tf7(QA-L9) = QA-P38.
QA-Tf7(L) = ConicPerspector[QA-Tf2(L)]. See QA-Co-1 and see Ref-34, QFG#2094.
 

 
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