QL-L6: Quasi Ortholine


The Quasi Ortholine originates from the centers from 3 parallelograms that are collinear on the Quasi Ortholine.
These parallelograms are each constructed from the 3 Quadrigons of the Reference Quadrilateral.
Let P1.P2.P3.P4 be a Quadrigon and S be the intersection point of the diagonals:
S = P1.P3 ^ P2.P4
h ij = Orthocenter in Triangle S.Pi.Pj (i and j consecutive nrs in the cycle 1,2,3,4)
Now h12.h23.h34.h41 is a parallelogram.
It is remarkable that:
line h41.h12 passes through P1,
line h12.h23 passes through P2,
line h23.h34 passes through P3,
line h34.h41 passes through P4.
The parallelogram has Center Ha = QG-P10 = 2nd Quasi Orthocenter.
Do this for all 3 Quadrigons in a Quadrilateral.
This gives centers Ha, Hb, Hc. These centers Ha, Hb, Hc are collinear.
The line through Ha, Hb, Hc also passes through QL-P2 (Morley Point) as well as QL-P10 (Orthocenter of the QL-Diagonal Triangle) in the Quadrilateral.
The line is called Quasi Ortholine because it passes through the 2nd Quasi Orthocenters of the 3 component Quadrigons of the Reference Quadrilateral.
 QL-L6-OrthoParLine-01
QL-L6-OrthoPar Line-00

Coefficients/Coordinates:
1st CT-coefficient:
 l m n (b2 (l - m) - c2 (l - n)) + l (m - n) (l m + l n - m n) SA
1st CT-Coordinate Infinity point:
                l m n (a2 (2 l - m - n) - (b2 - c2) (m - n)) - m (l - n) (l m - l n + m n) SB + n (l - m) (l m - l n - m n) SC
1st CT-Coordinate Infinity point of perpendicular line to QL-L6:              (very simple coordinates)
                l (m - n) (-3 m n + l m + l n)
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1st DT-coefficient:
            l2 (m2-n2) SA
1st DT-Coordinate Infinity point:
            m2 (l2 - n2) SB + n2 (l2 - m2) SC
1st DT-Coordinate Infinity point of perpendicular line to QL-L6:              (very simple coordinates)
            l2 (m2-n2)

Properties:
  • QL-L6 passes through QL-P2 (Morley point) and QL-P10 (Orthocenter QL-Diagonal Triangle)
  • QL-L6 is parallel to QL-L5 (NSM Line)
  • QL-L9 (M3D Line) is perpendicular to QL-L6.
  • The circumcenter of the triangle formed by the 3 QL-versions of QG-P13 lies on QL-L6.
  • The 3 QL-versions of QA-P14 lie on QL-L6 (note Eckart Schmidt).
  • The 3 QL-versions of QG-P10 lie on QL-L6 (note Eckart Schmidt).
  • QL-L6 is the Steiner Line (QL-L2) of the Quadrilateral formed by the lines of the QL-Diagonal Triangle (QL-Tr1) and the Newton Line (QL-L1). See Ref-34, message # 164 and accompanying document “Diagonal Quadrilaterals and Conics through the points S1,S2,S3” page 2 of Bernard Keizer.

 

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