QL-Co3: 2^nd QL-Parabola

The points of tangency of the inscribed QL-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 [43] and [34], QFG message # 504.

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

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 [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 QL’s QL1 and QL2 as well as their Newton Line are couples of Isotomic Transversals (see [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 [34], QFG#1315).

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