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 [31] as midpoint of QL-P2 and QL-P4 lying on QL-L1, the Newton Line.

1^st CT-coordinate

(m – n) (a² S_A l² + b² S_B l m + c² S_C l n + 8 Δ² (l m + l n – 2 m n – l²))

1^st 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-P22 lies on these lines:
  • QL-P2.QL-P4           (1 : 1 => QL-P22 is Midpoint of QL-P2 and QL-P4)
  • QL-P5.QL-P7           = Newton Line (QL-L1)
• QL-P22.QL-P6 // QL-P2.QL-P5 // QL-P4.QL-P20
• QL-P22 is the Midpoint of QL-P5 (Clawson Center) and QL-P20 (Orthocenter Homothetic Center).
• The distance ratios between points QL-P20, QL-P22, QL-P12, QL-P5 are 3 : 1 : 2.

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