QL-P27: Nine-point Homothetic Center QL-X(3) Quadrangle

QL-P27 is the QA-P7 point (Nine-point Center Homothetic Center) of the Circumcenter Quadrangle (see QA-1 and QL-P4) of the Reference Quadrilateral.

This point was found by Eckart Schmidt (November 26, 2012).

1^st CT-coordinate

2 a⁴ (m – n) (l – m) (l – n) + b² c² (m – n) (l m + l n – 2 m n)

+ b⁴ n (m – l) (m – n) + a² c² ( n – l ) (l m – 3 m² – 2 l n + 4 m n)

+ c⁴ m (n – m) (-n + l) + a² b² (m – l) (2 l m – l n – 4 m n + 3 n²)

1^st DT-coordinate

3 a⁴ (l² – m²) ( l² – n²) (m² – n² ) + 4 b² c² (m² – n²) (-2 m² n² + l² m² + l² n²)

– b⁴ (l² – m²) (m² – n²) (m² + 3 n²) + 2 a² c² ( l² – n²) (3 m⁴ – 5 m² n² + l² m² + l² n²)

– c⁴ (l² – n²) (m² – n²) (3 m² + n²) + 2 a² b² (m² – l²) (l² m² + l² n² – 5 m² n² + 3 n⁴)

Properties

• QL-P27 lies on these lines:
  • QL-P3.QL-P4 (2 : 1)
  • QL-P22.QL-P29
• QL-P27 is the Perspector of the Reference Quadrilateral and the 2nd generation X(5)-Quadrangle, where X(5) = Nine-point Center of a QL-Component Triangle.

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