QL-Tr2: QL-NPC-Triangle


The three QA-Ninepoint Conics (QA-Co1) in a Quadrilateral have 3 common points.
Per pair of QA-Ninepoint Conics there is a 4th intersection point that lies on the Newton Line QL-L1. The triangle formed by the 3 common points will be called here the NPC-Triangle (NinePointConic-Triangle). This triangle has several special properties. More information can be found at Ref-34, QFG#678 in an analyses by Eckart Schmidt, where all properties described below can be found, unless otherwise referenced. Bernard Keizer mentioned these points also at Ref-34, QFG#457 and #1458.
 QL-Tr2-NPC Triangle-01

CT-coordinates of the 3 vertices:
(A1              +          B1             +             C1   :   A2             +          B2              +            C2   :   A3)
(A1 - ½ (1  - i √3) B1 - ½ (1 + i √3]) C1   :   A2 - ½ (1 + i √3) B2 - ½ (1  - i √3]) C2   :   A3)
(A1 - ½ (1 + i √3) B1 - ½ (1  - i √3]) C1   :   A2 - ½ (1  - i √3) B2 - ½ (1 + i √3]) C2   :   A3)
where:
A1 =   3 m (l - n) (2 l2 - 4 l m - l n + 3 m n)
B1 = -3 m (l - n) U1
C1 =   3 m (l - n) U4/U1
A2 =   3  l  (l - n) (4 l m - 2 m2 - 3 l n + m n)
B2 = -(2 l2 - l m - 4 l n + 3 m n) U4/U1  +  U12 
C2 =   (2 l2 - l m - 4 l n + 3 m n) U1 + (U4/U1)2
A3 =   9 l m (l - m)(l - n) U1
where:
U1 =
(-8 l6 + 21 l5 m - 15 l4 m2 + 10 l3 m3 + 21 l5 n - 75 l4 m n + 75 l3 m2 n - 45 l2 m3 n - 15 l4 n2 + 75 l3 m n2 - 90 l2 m2 n2 + 54 l m3 n2 + 10 l3 n3 - 45 l2 m n3 + 54 l m2 n3 - 27 m3 n3 +
3 √(3(-l2 (l - m)2 (l - n)2 (9 l4 m2 - 9 l3 m3 + 9 l2 m4 - 14 l4 m n + 3 l3 m2 n + 3 l2 m3 n - 14 l m4 n + 9 l4 n2 + 3 l3 m n2 - 3 l2 m2 n2 + 3 l m3 n2 + 9 m4 n2 - 9 l3 n3 + 3 l2 m n3 + 3 l m2 n3 - 9 m3 n3 + 9 l2 n4 - 14 l m n4 + 9 m2 n4))))1/3
U4 = -4 l4 + 7 l3 m - 7 l2 m2 + 7 l3 n - 11 l2 m n + 12 l m2 n - 7 l2 n2 + 12 l m n2 - 9 m2 n2

Properties:
  • QL-P12 is the centroid of QL-Tr2.
  • QL-P2 is the orthocenter of QL-Tr2.
  • QL-P6 is the circumcenter of QL-Tr2.
  • Consequently is QL-P2.QL-P6 the Eulerline of QL-Tr2.
  • The vertices of QL-Tr2 lie on the Dimidium Circle QL-Ci6.
  • The sides of QL-Tr2 are tangents to the inscribed Quadrilateral Parabola QL-Co1. For an analyses of the triangle formed by the points of tangency see Ref-34, QFG-message #1460 by Eckart Schmidt.
  • The triangles QL-Tr1 and QL-Tr2 have a common circumconic containing QL-P8, QL-P13, QL-P24.
  • The Isogonal Conjugate of QL-P1 wrt QL-Tr2 is the infinity point of QL-L1.
  • The Isogonal Conjugate of QL-P17 wrt QL-Tr2 is the infinity point of QL-L9.
  • Every QL-Component Triangle has a conic Coi (i=1,2,3,4) through its vertices, centroid and its perspector with the QL-Diagonal Triangle. These 4 conics Coi (i=1,2,3,4) also have 3 common points being the vertices of QL-Tr2 (see Ref-34, Bernard Keizer, QFG#457 and #1458 also for more properties).