QL-P22: QL-Nine-point Center Homothetic Center


QL-P22 is the homothetic Center of the Reference Quadrilateral L1.L2.L3.L4 and the homothetic Quadrilateral formed by the lines parallel to the lines L1, L2, L3, L4 through the Nine-point Centers of the corresponding component triangles.
The prefix “QL-“ has been added in the name of QL-P22 because there is also a Nine-point Homothetic Center in a Quadrangle.
J.W. Clawson mentioned this point in 1921 in his document at Ref-31 as midpoint of QL-P2 and QL-P4 lying on QL-L1, the Newton Line.

QL-P22-QL-Ninepoint-Homothetic-Center               
Coordinates: 
1st CT-coordinate:
(m – n) (a2 SA l2 + b2 SB l m + c2 SC l n + 8 Δ2 (l m + l n - 2 m n - l2))
1st DT-coordinate:
(m2 - n2) ((m2 - n2)2 Sa2 + (n2 - l2)2 Sb2 + (l2 - m2)2 Sc2 - 2 S2 (3 l2 m2 + 3 l2 n2 - 5 m2 n2) + 2 (l4 Sb Sc + m4 Sa Sc + n4 Sa Sb))

Properties:
        QL-P2.QL-P4                     (1 : 1 => QL-P22 is Midpoint of QL-P2 and QL-P4)
        QL-P5.QL-P7                      = Newton Line (QL-L1)
 

 

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