QL-Co-1: Inscribed Quadrilateral Conics

5^th 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 5^th 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 5^th 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 5^th 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 )

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

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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