QL-Tf7: QL-5th Line touchpoint

QL-Tf7 maps a line L into the point of tangency of L at the inscribed conic of L1,L2,L3,L4,L. It is the dual of QA-Tf9.

Construction:
Definition of Triangle Transformation TR-Tfx:
TR-Tfx(ABC,L,Li) = L-Isoconjugate for lines wrt triangle ABC of Li.
It can be constructed as QL-Tf2(Li) wrt quadrilateral (La,Lb,Lc,L) , where La,Lb,Lc are the sidelines of the Anticevian Triangle of L wrt ABC (construction from Eckart Schmidt, see Ref-34, QFG#2157, #2164).
Let L_1, L_2, L_3, L_4 be the basic lines from the Reference Quadrilateral, also referred to as L_i (i=1,2,3,4). Let L be some random line. Let QL-DT = Diagonal Triangle QL-Tr1.
Then QL-Tf7(L) is the common intersection point of L and the 4 occurrences of L_i’ = TR-Tfx(QA-DT,P,Pi). CT-Coordinates:
Let L = (x:y:z), then QL-Tf7(L) =
(l y z (-n y + m z) : m x z (n x - l z) : n x y (-m x + l y))

Properties:
QL-Tf7(L) = L ^ QL-Tf2(L).
QL-Tf7(QL-L1) = Infinitypoint Newton Line
QL-Tf7(QL-L2) = QL-L2 ^ (Line through QL-P1 parallel to QL-P3.QL-P4)
QL-Tf7(QL-L3) = QL-L3 ^ (Line through QL-P1 parallel to Newton Line)
• Let P12=L1^L2, P13=L1^L3, P14=L1^L4, P23=L2^L3, P24=L2^L4, P34=L3^L4. Let S12, S13, S14, S23, S24, S34 be the projection points on L from P12, P13, P14, P23, P24, P34. Let L12, L13, L14, L23, L24, L34 be the connecting lines of P12 and S34, P13 and S24, P14 and S23, P23 and S14, P24 and S13, P34 and S12 respectively. Finally L12^L34, L13^L24 and L14^L23 are collinear on a line through QL-Tf7(L). For more properties see Ref-34, QFG#2215.