QG-P6: 1st QG-Quasi Orthocenter

The 1st QG-Quasi Circumcenter is the Diagonal Crosspoint of the X4-Quadrigon.
The X4-Quadrigon is defined by its vertices being the Triangle Orthocenters of the component triangles of the Reference Quadrigon.
This point and other 1st QG-Quasi points are described in Ref-5.

Coordinates:

CT-Coordinates QG-P6 in 3 QA-Quadrigons:           (only coordinates of 1st Quadrigon point are given)
•             (SB SC (p2 + r2) + (a2 - c2) SB p q + (a2 + SB) SC q r + SC (a2 + c2) p r   :
SA SC (p2 + r2) + (a2 - c2) SA p q - (a2 - c2) SC q r - (a4 + b4 - 2 a2 c2 + c4) p r/2   :
SA SB (p2 + r2) - (a2 - c2) SB q r + (c2 + SB) SA p q + SA (a2 + c2) p r)
CT-Coordinates QG-P6 in 3 QL-Quadrigons:           (only coordinates of 1st Quadrigon point are given)
• (c4 (m - n) (l2 m - l2 n - m2 n) - 2 b2 c2 (m - n) (l2 m - l m2 - l2 n + l m n - m2 n) + b4 (m - n) (l2 m - 2 l m2 - l2 n + 2 l m n - m2 n) + a4 (-l2 m2 - 2 l2 m n + 4 l m2 n - m3 n - l2 n2 + 2 l m n2 - m2 n2) + a2 (2 b2 m3 (l - n) - 2 c2 m n (2 l m - m2 - l n))   :
2 b2 c2 m (l m2 - l m n - m2 n + l n2) + b4 (-l2 m2 + 2 l m3 + 2 l2 m n - 5 l m2 n + 2 m3 n - l2 n2 + 2 l m n2 - m2 n2) + c4 (l2 m2 - 2 l2 m n - l m2 n + l2 n2 + m2 n2) + a4 (l2 m2 - l m2 n + l2 n2 - 2 l m n2 + m2 n2) + a2 (2 b2 m (-l m2 + l2 n - l m n + m2 n) - 2 c2 (l2 m2 - l2 m n - l m2 n + l2 n2 - l m n2 + m2 n2))   :
-2 b2 c2 m3 (l - n) + a4 (l - m) (l m2 + l n2 - m n2) + b4 (l - m) (l m2 - 2 l m n + 2 m2 n + l n2 - m n2) + c4 (-l2 m2 - l m3 + 2 l2 m n + 4 l m2 n - l2 n2 - 2 l m n2 - m2 n2) + a2 (2 c2 l m (m2 + l n - 2 m n) - 2 b2 (l - m) (l m2 - l m n + m2 n + l n2 - m n2)))
CT-Area of QG-P6-Triangle in the QA-environment:
• 2 p q r Δ / ((p + q) (p + r) (q + r))                                                         (equals area QA-Diagonal Triangle)
CT-Area of QG-P6-Triangle in the QL-environment:
•                4 l2 m2 n2 Δ / (( l m - l n - m n) (l m + l n - m n) (l m - l n + m n))                     (equals area QL-Diagonal Triangle)

DT-Coordinates QG-P6 in 3 QA-Quadrigons:           (only coordinates of 1st Quadrigon point are given)
•           (-2 (c2 p2-a2 r2) (Sc (-p2+q2+r2)+2 Sb q2)   :
-S2 (p2-q2-r2) (p2+q2-r2)-2 Sa2 p2 (p2+q2-r2)+2 Sc2 r2 (p2-q2-r2)-4 Sb q2 (Sa p2+Sc r2)   :
2 (Sa (p2+q2-r2)+2 Sb q2) (c2 p2-a2 r2))
DT-Coordinates QG-P6 in 3 QL-Quadrigons:           (only coordinates of 1st Quadrigon point are given)
•               (-2 Sb m2 (Sb (l2-n2)+Sc (l2-m2))   :
-S2 (m2 (l2+m2+n2)-l2 n2)+2 m2 (Sb Sc l2+Sa Sc m2+Sa Sb n2)   :
2 Sb m2 (Sb (l2-n2)+Sa (m2-n2)))
DT-Area of QG-P6-Triangle in the QA-environment:
•         S/2                                                                                      (equals area QA-Diagonal Triangle)

DT-Area of QG-P6-Triangle in the QL-environment:
•         S/2                                                                                      (equals area QL-Diagonal Triangle)

Properties:

• QG-P4, QG-P5, QG-P6, QG-P7 are collinear on QG-L4, the 1st QG-Quasi Euler Line.
• QG-P6 is the Orthocenter of the 1st QG-Quasi Diagonal Triangle: QG-Tr1.
• QA-P2 (Euler-Poncelet Point) lies on the circumcircle of the triangle formed by the 3 QA-versions of QG-P6.
• The area of the triangle formed by the 1st QG-Quasi Orthocenters of the 3 QA-Quadrigons equals the area of the QA-Diagonal Triangle.
• The area of the triangle formed by the 1st QG-Quasi Orthocenters of the 3 QL-Quadrigons equals the area of the QL-Diagonal Triangle.
• The triangle formed by the 1st QG-Quasi Orthocenters of the 3 QL-Quadrigons is perspective with the QL-Diagonal Triangle. The Perspector is the infinity point of QL-L2 (Steiner Line).
• QG-P6 is the Reflection of QG-P1 in QG-P10.
• The line through the 1st and 2nd QG-Quasi Orthocenters (QG-P6 and QG-P10) is perpendicular to the Newton Line (QL-L1).

Vernieuwen