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).

Coordinates:

1st 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)
1st 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-P3.QL-P4              (2 : 1)
• 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.

Vernieuwen