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QG-P6: 1st Quasi Orthocenter

The 1st 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 Quasi points are described in [5].

Infovisual QG-P6-infovisual-cvt-01.png
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 Quasi Orthocenters of the 3 QA-Quadrigons equals the area of the QA-Diagonal Triangle.
  • The area of the triangle formed by the 1st Quasi Orthocenters of the 3 QL-Quadrigons equals the area of the QL-Diagonal Triangle.
  • The triangle formed by the 1st 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 Quasi Orthocenters (QG-P6 and QG-P10) is perpendicular to the Newton Line (QL-L1).


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