QLCu1: QLQuasi Isogonal Cubic
QLCu1 has several special characteristics:
 QLCu1 is the locus of all points P for which the Isogonal Conjugates wrt all 4 component triangles coincide in a point on the same cubic. So the cubic is in terms of a quadrilateral QLselfisogonal.
 QLCu1 is the locus of all points P for which the QGQuasi Isogonal Conjugates wrt all 3 component quadrigons coincide in a point on the same cubic.
 QLCu1 is the locus of the foci of all inscribed conics in the Reference Quadrilateral.
 QLCu1 is the locus of all points for whom the Midpoint of this point and its ClawsonSchmidt Conjugate lie on the Newton Line QLL1.
 QLCu1 is the locus of all points P for which the reflections of P in the 4 quadrilateral lines are concyclic (Seiichi Kirikami in Ref34, EQF message #1093).
 QLCu1 is the locus of all points P for which the feet of the perpendiculars from P to the 4 quadrilateral lines are concyclic (all lie on a circle).

QLCu1 is the locus of points, whose Antipedal Quadrigon is cyclic (if not degenerated) (see Ref34, message #355 from Eckart Schmidt).
 QLCu1 is the locus of all points P where all perpendiculars from Li^Lj at Si.Sj coincide in a point on the same cubic, where Si = pedal point of P on Li.

QLCu1 is the locus of points, whose reflections in the circumcircles of the QLComponent Triangles are concyclic (Eckart Schmidt in Ref34, EQF message #403).
If the Reference System is the Reference Quadrilateral, this cubic is a circular isocubic invariant to the ClawsonSchmidt Conjugate QLTf1.
If the Reference System is the Orthic Triangle of the QLDiagonal Triangle, this cubic is an isogonal circular cubic.
This Quadrilateral Cubic is also described (with other names) by Fred Lang (see Ref24, page 3) and Eckart Schmidt (see Ref15e].
More information about the type of cubic is described by Bernard Gibert at Ref17b and Ref17d.
More information about the subject also can be found at Ref43.
How to draw a line through
all 6 points of a Quadrilateral . . .
Construction:
 Let P be some point on QLL2 (Steiner Line).
 P* = Isogonal Conjugate of P wrt Triangle L2.L3.L4 (or any other QLComponent Triangle).
 Co1= Conic through vertices triangle L2.L3.L4 and QLP1 and P*.
 Pr = P reflected in QLL1 (Newton Line).
 Lr = line through Pr // QLL1 (now P is railway watcher).
 S1 and S2 are intersection points Lr ^ Co1.
Conjugates on QLCu1
It is special that this conjugate has the same performance for points on QLCu1 as:
 the Isogonal Conjugate of P wrt the Orthic Triangle of the QLDiagonal Triangle (note Eckart Schmidt).
 the Quasi Isogonal Conjugate (QGTf2) of P wrt any Quadrigon of the Reference Quadrilateral (note Eckart Schmidt).
Other properties of this conjugate:
Easy construction method of ClawsonSchmidt conjugated points on QLCu1
Any point X on QLCu1 can be seen as the intersection point of the line Ln parallel to the Newton line and the connecting line Lp=QLP1.X.
Now CSC(X) can be constructed as the intersection point of La’ and Lb’, where:
Ln’ = Ln Reflected in the Newton Line
Lp’ = Lp reflected in the 1^{st} Steiner Axis (see description at QLTf1)
The 1st Steiner Axis easily can be constructed as:
As a consequence, when X1 and X2 are CSconjugated points, then the line through X1 parallel to QLL1 and the line X2.QLP1 will cross in another point on QLCu1. X1 and X2 can be interchanged here.
Two interesting new points on QLCu1
Tangents at QLCu1
 The tangent at QLCu1 in QLP1 = QLP1.QLP4. This line is also tangent in QLP1 at the circle (QLP1,QL2P2a,QL2P2b), which is the ClawsonSchmidt Conjugate (QLTf1) of the Newton Line QLL1.
 The tangent at QLCu1 in T is S.T reflected in the perpendicular bisector of line segment QL2P2a.QL2P2b.
General construction method of the tangent in a point P on QLCu1
 Let P* be the isogonal conjugate of P.
 Let Co1 the isogonal conjugate of PP* wrt triangle L1,L2,L3.
 Let Co2 the isogonal conjugate of PP* wrt triangle L2,L3,L4.

Let Q be the 4^{th} intersection point of Co1 and Co 2.

Then PQ is the tangent in P wrt QLCu1.
Equations/Coordinates:
Equation in CTnotation:
a^{2} l (m y + n z) y z + b^{2} m (n z + l x) x z + c^{2} n (l x + m y) x y
+ 2 (S_{A} m n + S_{B} l n + S_{C} l m) x y z = 0
CTcoordinates Infinity Point Asymptote:
( l (m  n) : m (n  l) : n (l  m) )
Equation in DTnotation:
l^{2} x^{2} (SA x – SB y – SC z)– m^{2} y^{2} (SA x – SB y + SC z) – n^{2} z^{2} (SA x + SB y – SC z)
– (a^{2} l^{2 }+ b^{2} m^{2 }+ c^{2} n^{2}) x y z = 0
DTcoordinates Infinity Point Asymptote:
( m^{2} – n^{2} : n^{2} – l^{2} : l^{2} – m^{2} )
Properties:

These points lie on QLCu1:
– all 6 intersection points of the 4 lines of the Reference Quadrilateral
– QLP1: the Miquel Point
– QGP17 a/b/c: the vertices of the Orthic Triangle of the QLDiagonal Triangle
– the 3 QLversions of QG2P5a and QG2P5b: the intersection points of QGCi2 with the QGDiagonals
– there are exactly 2 points on the Newton Line that are each other’s Isogonal Conjugate wrt all 4 QLComponent Triangles. Both points lie on QLCu1. See Ref34, QFG #179.
– The Trilinear Poles of Li wrt triangle Lj.Lk.Ll, where (i,j,k,l) ∈ (1,2,3,4).
– the intersection point T of the perpendicular bisector of line segment QL2P2a.QL2P2b and the line parallel to QLL1 through QLP1.
(S and T are a QLTf1conjugated pair of points)
– the circular points at infinity (so the cubic is called circular, see Ref17c)
 The tangents at the circular points at infinity meet at the Miquel Point QLP1. This point also lies on QLCu1, which makes this cubic a Van Rees Cubic / Focal Cubic. See Ref17c.
 The line QLP1.QLP4 is tangent at QLCu1 in QLP1.
 The asymptote of QLCu1 // QLL1 = Newton Line.
 Distance Miquel Point (QLP1) to Asymptote is twice the distance from Miquel Point to Newton Line (Railway Watcher system, see QLL1).
 QLP21 lies on the asymptote of QLCu1.
 The ClawsonSchmidt Conjugate (QLTf1) of some point P on the cubic is also a point on the cubic. Thus the cubic is invariant under ClawsonSchmidtconjugation. See Ref15e.
 The ClawsonSchmidt Conjugate (QLTf1) of some point P on QLCu1 is identical with the Isogonal Conjugate of P wrt the Orthic Triangle of the QLDiagonal Triangle (note Eckart Schmidt).
 The ClawsonSchmidt Conjugate (QLTf1) of some point P on QLCu1 is identical with the Quasi Isogonal Conjugate (QGTf2) of P wrt any Quadrigon of the Reference Quadrilateral (note Eckart Schmidt).
 QLCu1 is the locus of intersection points of QGTf2(L) and QLTf1(L), where L = variable line through QGP1 (Ref34, Eckart Schmidt, QFG message #224).
 The circle through QL2P2a & QL2P2b and QLP1 is the QLTf1 image of the Newton Line QLL1 and is tangent at QLCu1 in QLP1. The common tangent at QLP1 is the line QLP1.QLP4.

QLCu1 is the locus for points whose angle bisectors wrt two opposite vertices of the Reference Quadrilateral are QLTf2 partners (Eckart Schmidt, Ref34, QFG message #1175).
 For any fixed point X on QLCu1, for any pair of variable QLTf1 conjugate points Y and Y' (including obviously the opposite vertices of the QL), the direction of the bisectors of the angle YXY' is the same (Eckart Schmidt, Bernard Keizer, see Ref34, QFG#1206, #1207).
 Let A, B, C, QLP1 be concyclic points on QLCu1, then QLCu1 is invariant wrt the ABCisogonal conjugate (Eckart Schmidt, see Ref34, QFG#1399).