QG-Tr1: 1st QG-Quasi Diagonal Triangle


QG-Tr1 is the Triangle with vertices QG-P1 (Diagonal Crosspoint) and the Reflections of QG-P1 in the 2 Diagonal Midpoints of the Reference Quadrigon.
Special about this triangle is that the 1st Quasi points QG-P4, QG-P5, QG-P6, QG-P7 are the corresponding triangle points of this Triangle.
For example the 1st Quasi Centroid QG-P4 of the Reference Quadrigon is also the Triangle Centroid of QG-Tr1.
This means that the Centroid of a Quadrigon also can be constructed as the Centroid of the 1st QG-Quasi Diagonal Triangle.
QG-Tr1-1st-QuasiDiagonalTriangle-01

Areas:
 
Area QG-1st Diagonal Triangle in 3 QA-Quadrigons CT-notation:
  • (r - p) (p + 2 q + r) S / (2 (r + p) (p + q + r))
  • (q - r) (2 p + q + r) S / (2 (q + r) (p + q + r))
  • (p - q) (p + q + 2 r) S / (2 (p + q) (p + q + r))
Area QG-1st Diagonal Triangle in 3 QL-Quadrigons CT-notation:
  • (-l m + l n + m n) (l m + l n - m n) S / (2 (m - l) (m - n) ( l m - l n + m n))
  • ( l m - l n + m n) (l m + l n - m n) S / (2 (n - l) (n - m) (-l m + l n + m n))
  • (-l m + l n + m n) (l m - l n + m n) S / (2 (l - m) ( l - n) ( l m + l n - m n))
 

Area QG-1st Diagonal Triangle in 3 QA-Quadrigons DT-notation:
  • 4 p r (p - r) (p + r) S / ((p + q + r) (-p + q + r) (p + q - r) (p - q + r))
  • 4 q r (q - r) (q + r) S / ((p + q + r) (-p + q + r) (p + q - r) (p - q + r))
  • 4 p q (q - p) (q + p) S / ((p + q + r) (-p + q + r) (p + q - r) (p - q + r))
Area QG-1st Diagonal Triangle in 3 QL-Quadrigons DT-notation:
  • 2 m4 S / ((m2 - l2) (m2 - n2))
  • 2 n4 S / ((n2 - l2) (n2 - m2))
  • 2 l4 S / ((l2 - m2) (l2 - n2))



Properties:
  • The 1st QG-Quasi Euler line QG-L4 is also the Triangle Euler line of QG-Tr1.
  • The points 1st Quasi Centroid/Circumcenter/Orthocenter/Nine-point Center (resp. QG-P4, QG-P5, QG-P6, QG-P7) are also the Centroid/Circumcenter/ Orthocenter/Nine-point Center of QG-Tr1.