QL-Co2: Inscribed Midline Hyperbola

The Inscribed Midline Hyperbola is the conic tangent at the 4 basic lines of the Reference Quadrilateral and the Newton Line (or Midline). It is remarkable that the point of tangency at the Newton Line is at its infinity point. That makes the Newton Line the asymptote of this conic and this makes this conic a hyperbola.
 
 
QL-Co2-MidlineConic-00

Equations:
 
Equation Conic CT-notation:
MN2 x2 + NL2 y2 + LM2 z2 - 2 MN NL x y - 2 LM NL y z - 2 LM MN z x = 0
Coordinates Center of Conic in CT-notation:
     (   LM + NL  :    LM + MN     :    MN + NL   )                 (1st  presentation)
     (l (m - n) LL :  m (n - l) MM :  n (l - m) NN)                 (2nd presentation)
Coefficients 2nd asymptote in CT-notation : 
(l m + l n - m n)/LL : (l m - l n + m n)/MM : (-l m + l n + m n)/NN
 
where:
               MN = l (m – n) (l m + l n – m n)2
               NL  = m (n – l) (l m – l n + m n)2
               LM  = n (l – m) (l m – l n – m n)2
LL   = (l m + l n + m n)2 - 4 m n (l2 + m n)
MM = (l m + l n + m n)2 - 4 n l (m2 + n l)
NN  = (l m + l n + m n)2 – 4 l m (n2 + l m)
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Equation Conic DT-notation:
            l2 x2 / (m2 - n2) + m2 y2 / (n2 - l2) + n2 z2 / (l2 - m2) = 0
Coordinates Center of Conic in DT-notation:
            ( m2 n2 (m2-n2)  :   n2  l2 (n2- l2)   :   l2  m2 (l2-m2))
Coefficients 2nd asymptote in DT-notation :
            (l2 (-l2+m2+n2)  :  m2 (l2-m2+n2)  :  n2 (l2+m2-n2))

Properties:

  • QL-Co2 is the 5th Line Conic of QL-L1 (see QL-Co-1), which is an asymptote.
  • The Center of the conic is QL-P23. This is a point on the Newton Line QL-L1, like all centers of inscribed Quadrilateral Conics are.
  • Algebraically this conic is independent of (a,b,c) just like the Newton Line.
  • QL-P13 lies on the Polar (see Ref-13, Polar) of QL-P12 wrt QL-Co2 and vice versa.