QL-L9: M3D Line


In the Reference Quadrilateral 3 QL-versions of the M3D Hyperbola QG-Co3 can be constructed. These 3 hyperbolas have pairwise a 3rd intersection point (the 1st and 2nd intersection point are mutual intersection points of L1, L2, L3, L4). These three 3rd intersection points coincide with the 3 QL-versions of QG-P15. They are collinear on a line which is called here the M3D Line.
M3D stands for “Midpoint 3rd Diagonal”.

Construction:
QL-L9 is the line through QL-P18 (the Reflection of QL-P8 in QL-P12) perpendicular to QL-L6 (Quasi Ortholine).
QL-L9-M3D-Line

Coeffficients/Coordinates:
1st CT-coefficient:
            m n (2 l2 + m n - l m - l n)
1st CT-Coordinate Infinity point:
l (m - n) (3 m n - l m - l n)
1st DT-coefficient:
            l2 (m2+n2)
1st DT-Coordinate Infinity point:
            l2 (m2-n2)

Properties:
  • QL-P18 and QL-P23 lie on this line.
  • QL-L9 is perpendicular to the lines QL-L5 (NSM Line) and QL-L6 (Quasi Ortholine).
  • The axis of the 2nd QL-Parabola QL-Co3 // QL-L9.
  • d(QL-P8, QL-L9) = 2 * d(QL-P12, QL-L9) =4/3 * d(QL-P19, QL-L9)
  • The 3 QL-versions of QG-P15 are collinear on QL-L9.
  • The 3 QA-versions of QL-L9 coincide in QA-P27.
  • The axes of the 2 circumscribed QA-parabolas (see QA-2Co1) of the Quadrangle formed by the centroids of the 4 Component Triangles of the Reference Quadrilateral are parallel to QL-L9 and QL-L1 (Eckart Schmidt, September 18, 2012).