QL-Co3: 2nd Quadrilateral Parabola


The points of tangency of the inscribed Quadrilateral Parabola QL-Co1 form a Quadrangle T1.T2.T3.T4. This Quadrangle has 2 circumscribed parabola's where QL-Co1 is one of them. The other circumscribed parabola is QL-Co3.

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

 
QL-Co3-2nd-Parabola-00
Equations:  
Equation in CT-notation:
    l2 (m - n)2 ((l m + l n + m n)2 - 8 l2 m n) x2 + 2 m n (l - m) (l - n) (l m + l n - m n)2 y z
+ m2 (l - n)2 ((l m + l n + m n)2 - 8 l m2 n) y2 + 2 l n (m - l) (m - n) (l m - l n + m n)2 x z
+ n2 (l - m)2 ((l m + l n + m n)2 - 8 l m n2) z2 + 2 l m (n - l) (n - m) (l m - l n  - m n)2 x y = 0
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Equation in DT-notation:
            m2 n2 (l2 - m2)(l2 - n2) x2 + l2 n2 (m2 - l2)(m2 - n2)y2 + l2 m2 (n2 - l2)(n2 - m2)z2 = 0

Properties:
  • The focus of QL-Co3 is QL-P25 which is the complement of QL-P17 wrt the QL-Diagonal Triangle.
  • The axis of the 2nd QL-Parabola // QL-L9 (M3D Line through QL-P18, QL-P23).
  • QL-P1.QL-P7 _|_ Axis QL-Co3, as a consequence QL-P1.QL-P7 // Directrix QL-Co3.
  • Directrices QL-Co1 and QL-Co3 meet in QL-P9 (Circumcenter QL-Diagonal Triangle).
  • QL-P25 lies on the Polar (see Ref-13, Polar) of QL-P9 wrt QL-Co3 and vice versa.
  • QL-P11 lies on perpendicular bisector F1.F2 (F1, F2 are Foci QL-Co1, QL-Co3).
  • The tangents at T1,T2,T3,T4 of QL-Co3 form a tangential quadrilateral. The QL-DT of this new quadrilateral (QL2) as well as the QL-DT of the Reference Quadrilateral (QL1) as well as the QA -DT of T1.T2.T3.T4 are identical.
  • The 4 lines forming the Quadrilaterals QL1 and QL2 as well as their Newton Line are couples of Isotomic Transversals (see Ref-13) wrt QL-DT. The 4 perspectors of the Component Triangles wrt QL-DT as well as QL-P13 of QL1 and QL2 are Isotomic Conjugates wrt QL-DT (Bernard Keizer, see Ref-34, QFG#1315).