5th Line Conics
By adding an extra line to the set of 4 basic lines of a Quadrilateral we get a configuration of 5 lines. Let L5 (u:v:w) be a random 5th line. Five lines do define a unique (inscribed) conic just like 5 points do define a unique (circumscribed) conic.
Equation Conic in CT-notation:
Tx2 x2 + Ty2 y2 + Tz2 z2 + 2 Ty Tz y z + 2 Tz Tx x z + 2 Tx Ty x y = 0
where:
Tx = l u (m w - n v)
Ty = m v (n u - l w)
Tz = n w ( l v - m u)
CT-Coordinates Center:
(Ty + Tz : Tx + Tz : Tx + Ty)
CT-Coefficients Asymptotes:
Asy-1    (Tx (Tx + Ty) (Tx + Ty + Tz) - (Tx + Ty) [-Tx Ty Tz (Tx + Ty + Tz)] :
-Ty(Tx + Ty) (Tx + Ty + Tz) - (Tx + Ty) [-Tx Ty Tz (Tx + Ty + Tz)] :
-Tz(Tx - Ty) (Tx + Ty + Tz) + (Tx + Ty + 2 Tz) [-Tx Ty Tz (Tx + Ty + Tz)] )
Asy-2    (Tx (Tx + Ty) (Tx + Ty + Tz) + (Tx + Ty) [-Tx Ty Tz (Tx + Ty + Tz)] :
-Ty(Tx + Ty) (Tx + Ty + Tz) + (Tx + Ty) [-Tx Ty Tz (Tx + Ty + Tz)] :
-Tz(Tx - Ty) (Tx + Ty + Tz) - (Tx + Ty + 2 Tz) [-Tx Ty Tz (Tx + Ty + Tz)] )

Equation Conic in DT-notation:
Ty Tz x2 + Tz Tx y2 + Tx Ty z2 = 0
where:
Tx = n2 v2 - m2 w2           Ty = l2 w2 - n2 u2           Tz = m2 u2 - l2 v2
DT-Coordinates Center:
(Tx : Ty : Tz)
DT-Coefficients Asymptotes:
Asy-1  (Tz (Ty2 + Tx Ty + Ty Tz - [-Tx Ty Tz (Tx + Ty + Tz)] :
-Tz (Tx2 + Tx Ty + Tx Tz + [-Tx Ty Tz (Tx + Ty + Tz)] :
(Tx + Ty) [-Tx Ty Tz (Tx + Ty + Tz)])
Asy-2  (Tz (Ty2 + Tx Ty + Ty Tz + [-Tx Ty Tz (Tx + Ty + Tz)] :
-Tz (Tx2 + Tx Ty + Tx Tz) - [-Tx Ty Tz (Tx + Ty + Tz)] :
-(Tx + Ty) [-Tx Ty Tz (Tx + Ty + Tz)])
Properties:
• QL-Co1 (Inscribed Parabola) and QL-Co2 (Inscribed Midline Hyperbola) are both examples of 5th Line Conics.
• The Center of a 5th Line Conic is always a point on the Newton Line QL-L1.

Center constructed inscribed Conics
It is also possible to define an inscribed Quadrilateral Conic by the 4 basic lines of the Reference Quadrilateral and the Center of the conic. This appointed center should be a point on the Newton Line (see properties 5th Line Conics).
Let N = (d : e : f) be a point on the Newton Line and be appointed as the center of an inscribed quadrilateral conic.
Equation Conic in CT-notation:
(-d+e+f)2 x2 - 2 ( d+e–f) (d–e+f) y z
+ (d-e+f)2 y2  - 2 (-d+e+f)(d+e–f) x z
+ (d+e–f)2 z2 - 2 (-d+e+f)(d–e+f) x y = 0
4 Points of tangency in CT-notation:
• (       0     : d+e–f : d–e+f )
• ( d+e–f :      0     : -d+e+f )
• ( d–e+f : -d+e+f :        0   )
• ( (d+e–f)2 m2+2 (d–e+f) f n2 + 2(d+e–f) (d–e+f) m n  :  (d+e–f)2 l2 +2 (-d+e+f) f n2 + 2(-d+e+f) (d+e–f) l n :  (d+e–f) ((d–e+f) l2+(-d+e+f) m2) )
Equation Conic in DT-notation:
e f x2 + d f y2 + d e z2 = 0
4 Points of tangency in DT-notation:
• ( - d l :   e m :   f n )
• (   d l : - e m :   f n )
• (   d l :   e m : - f n )
• (   d l :   e m :   f n )

Properties:
• When N = QL-L1 ^ QL-L6 then one of the axes of the Center Conic coincides with the Newton Line (note Eckart Schmidt).

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