**QL-Tf2: QL-Line Isoconjugate**

*Let a line intersect the diagonals of a quadrilateral, then the 4*

^{th}harmonic points on the diagonals are collinear.This Line Transformation as well as its properties were further described and elaborated by Eckart Schmidt (December 16, 2012). For these and further properties see below and Ref-34, QFG#1175.

*Construction:*

*Coefficients:**CT- coefficients:*

L(e : f : g) --> (e (-e m n + f l n + g l m) : f (e m n - f l n + g l m): g (e m n + f l n - g l m))

*DT- coefficients:*

L(e : f : g) --> ( f g l

^{2}: g e m^{2}: e f n^{2})

*Properties:*All not otherwise referenced properties are from Eckart Schmidt (mail December 16, 2012 and Ref-34, QFG messages #481, #1175).

- QL-L1 will be transformed in the infinity line.
- QL-L2 will be transformed in a parallel to QL-P3.QL-P4 through QL-P1.
- For a Quadrigon the QL-Tf2 transformation of QG-P2.QG-P13 will be the reflection of QG-L1 in QG-P1.
- The Steiner Axes (described at QL-Tf1) are QL-Tf2 partners.
- The QL-Tf2 images of lines through a fixed point envelope an inscribed conic of the QL-Diagonal Triangle QL-Tr1. For QL-P13 we get the inscribed Steiner Elli[se of QL-Tr1. For points on the Newton line these conics are inscribed parabolas of QL-Tr1. For the Miquel Point QL-P1 we get a special inscribed conic which contacts the Steiner Axes (described at QL-Tf1) and the line QL-L2 (see Ref-34, QFG#481.
- The intersections of lines through a fixed point Q and their QL-Tf2 images give a cubic through the six vertices of the reference Quadrilateral as well as the vertices of the Ceva Triangle of Q wrt QL-Tr1.
- QL-Cu1 is the locus for points whose angle bisectors wrt any two opposite vertices of the Reference Quadrilateral are QL-Tf2 partners.
- The intersections for QL-Tf2 images of perpendicular lines of a pencil are collinear.
- QL-Tf2(L)=Tripolar(QA-Tf2h(Tripole(L))), where QA-Tf2h=QA-Tf2 wrt the Quadrangle formed by the QL-Tr1-trilinear poles of the QL-defining lines (See Ref-34, QFG#1497,#1506 by Eckart Schmidt).
- The QL-Tf2-image of the QL-Co1-polar of a point P contains P. See Ref-34, Eckart Schmidt, QFG-message #1666.