QL-Co1: Inscribed Parabola


The inscribed Parabola is the unique parabola that can be inscribed within a Quadrilateral.

This parabola is also described by Bernard Keizer in Ref-43 and Ref-34, QFG message # 504.

 
QL-Co1-ParabolaInscribed-00
 
Equations/Infinity Points:
Equation CT-notation:
                l2 (m - n)2 x2 – 2 m n (l - m)(n - l ) y z
+ m2 (l – n)2 y2 – 2 n l (l - m)(m - n) x z
+ n(l - m)2 z2 – 2 l m (n - l)(m - n) x y = o
1st CT-coefficient Axis Parabola:
                l (m – n) (c l m + b l n – b m n – c m n) (c l m – b l n + b m n – c m n)
Infinity point Axis CT-notation:
(     l (m–n)  :   m (n – l)  :   n (l – m))
Points of tangency with L1, L2 , L3, L4 in CT-notation:
            (         0          :    n (l - m)  :      m (n - l))
            (     n (l - m)  :         0         :      l (m - n))
            (    m (n - l)   :    l (m - n)  :           0       )
            (m n (m - n) : n l (n  - l )  :  l m (l - m))
----------------------------------------------------------------------------------------------------------------
Equation DT-notation:
            x2/(m2-n2) + y2/(-l2+n2) + z2/(l2-m2) = 0
1st DT-coefficient Axis Parabola:
            (l2-m2) (l2-n2) ((l2-n2) SB+(l2-m2) SC)
Infinity point Axis DT-notation:
            ( m2- n2 : n2- l2 : l2 - m2 )
Points of tangency with L1, L2 , L3, L4 in DT-notation:
            (-l (m2 - n2) :   m (l2 - n2) :    (l2 - m2) n)
            (  l (m2 - n2) :  -m (l2 - n2) :   (l2 - m2) n)
            (  l (m2 - n2) :   m (l2 - n2) :  -(l2 - m2) n)
            (  l (m2 - n2) :   m (l2 - n2) :    (l2 - m2) n)

Properties:
  • QL-Co1 is the only parabola inscribed in the quadrilateral (see Ref-4, page 51).
  • QL-Co1 is the 5th Line Conic of QL-L3 (see QL-Co-1).
  • The Focus is QL-P1 the Miquel Point.
  • The Directrix is QL-L2 the Steiner Line.
  • The Axis of QL-Co1 is parallel to QL-L1, the Newton Line.
  • QL-Co1 is also an inscribed parabola of the QL-Medial Triangle (the medial triangle of the QL-Diagonal Triangle).
  • The Nine-point Conics of the 3 Quadrigons of a Quadrilateral have 3 common points which constitute triangle QL-Tr2. The Inscribed Parabola of the quadrilateral is also an inscribed parabola of QL-Tr2.
  • QL-P1 lies on the Polars (see Ref-13, Polar) of QL-P2, QL-P7, QL-P9 wrt QL-Co1.
  • QL-P8 lies on the Polar (see Ref-13, Polar) of QL-P12 wrt QL-Co1 and vice versa.
  • Let T1, T2, T3, T4 be the points of tangency of QA-Co1 with the basic lines of the Reference Quadrilateral. Now the QL-Diagonal Triangle of the Reference Quadrilateral and the QA-Diagonal Triangle of the Quadrangle T1.T2.T3.T4 are identical.
  • The QL-Tf2-image of the QL-Co1-polar of a point P contains P. See Ref-34, Eckart Schmidt, QFG-message #1666.