QG-Tr2: 2nd QG-Quasi Diagonal Triangle

 
QG-Tr2 is the Triangle with vertices QG-P1 (Diagonal Crosspoint) and the midpoints of the 2 diagonals of the Reference Quadrigon.
Special about this triangle is that the 2nd Quasi points QG-P8, QG-P9, QG-P10, QG-P11 are the corresponding triangle points of this Triangle.
For example the 2nd Quasi Centroid QG-P8 of the Reference Quadrigon is also the Triangle Centroid of QG-Tr2.
 
QG-Tr2-2nd-QuasiDiagonalTriangle-01

Areas:
        
Area QG-2nd Diagonal Triangle in 3 QA-Quadrigons CT-notation:
  • (r - p) (p + 2 q + r) S / (8 (p + r) (p + q + r))
  • (r - q) (2 p + q + r) S / (8 (q + r) (p + q + r))
  • (p - q) (p + q + 2 r) S/ (8 (p + q) (p + q + r))
 
Area QG-2nd Diagonal Triangle in 3 QL-Quadrigons CT-notation:
  • (-l m + l n + m n) (l m + l n - m n) S / (8 (m - l) (m - n) ( l m - l n + m n))
  • ( l m - l n + m n) (l m + l n - m n) S / (8 (n - l) (n - m) (-l m + l n + m n))
  • (-l m + l n + m n) (l m - l n + m n) S / (8 (l - m) ( l - n) ( l m + l n - m n))

Area QG-2nd Diagonal Triangle in 3 QA-Quadrigons DT-notation:
  • p r (p - r) (p + r) S / ((p + q + r) (-p + q + r) (p + q - r) (p - q + r))
  • q r (q - r) (q + r) S / ((p + q + r) (-p + q + r) (p + q - r) (p - q + r))
  • p q (q - p) (q + p) S / ((p + q + r) (-p + q + r) (p + q - r) (p - q + r))
 
Area QG-2nd Diagonal Triangle in 3 QL-Quadrigons DT-notation:
  • m4 S / (2 (m2 - l2) (m2 - n2))
  • n4 S / (2 ( n2 - l2) (n2 - m2))
  • l4 S / (2 ( l2 - m2) ( l2 - n2))
 
Properties:
 
  • The 2nd QG-Quasi Euler line QG-L5 is also the Triangle Euler line of QG-Tr2.
  • The points 2nd Quasi Centroid/Circumcenter/Orthocenter/Nine-point Center (resp. QG-P8, QG-P9, QG-P10, QG-P11) are also the Centroid/Circumcenter/ Orthocenter/Nine-point Center of QG-Tr2.
  • When the Reference Quadrigon is cyclic the Maltitudes (a Maltitude is a line drawn from the midpoint of a side perpendicular to the opposite side of a Quadrigon) concur in the orthocenter QG-P10 of QG-Tr2. This point in a cyclic Quadrigon is also known as the Anticenter.

 

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