QG-P8: 2nd QG-Quasi Centroid

 
Let P1.P2.P3.P4 be a Quadrigon and let S be its Diagonal Crosspoint QG-P1.
Let Gi i+1 = Centroid of Triangle Pi.Pi+1.S (i = cyclic sequence 1,2,3,4) .
Now is G12.G23.G34.G41 a parallelogram.
The sides of this parallelogram are parallel to the diagonals of the Reference Quadrigon.
QG-P8 is the Center of this parallelogram.
 
QG-P8-2ndQuasiCentroid-01

Coordinates:
 
CT-Coordinates QG-P8 in 3 QA-Quadrigons:              (only coordinates of 1st Quadrigon point are given)
  • (-12 p6 - 9 p5 q - 63 p5 r - 39 p4 q r - 135 p4 r2 - 66 p3 q r2 - 150 p3 r3 - 54 p2 q r3 - 90 p2 r4 - 21 p q r4 - 27 p r5 - 3 q r5 - 3 r6 : -3 p6 - 6 p5 q - 18 p5 r - 30 p4 q r - 45 p4 r2 - 60 p3 q r2 - 60 p3 r3 - 60 p2 q r3 - 45 p2 r4 - 30 p q r4 - 18 p r5 - 6 q r5 - 3 r6 : -3 p6 - 3 p5 q - 27 p5 r - 21 p4 q r - 90 p4 r2 - 54 p3 q r2 - 150 p3 r3 - 66 p2 q r3 - 135 p2 r4 - 39 p q r4 - 63 p r5 - 9 q r5 - 12 r6)
 
CT-Coordinates QG-P8 in 3 QL-Quadrigons:               (only coordinates of 1st Quadrigon point are given)
  • ((m - n) (-l2 m + 2 l m2 + l2 n - 5 l m n + 4 m2 n) :
-l2 m2 + 5 l2 m n - 4 l m2 n - 4 l2 n2 + 5 l m n2 - m2 n2 :
-(l - m) (4 l m2 - 5 l m n + 2 m2 n + l n2 - m n2))
 
CT-Area of QG-P8-Triangle in the QA-environment:        (equals 1/9 area QA-Diagonal Triangle)
  • 2 p q r Δ / (9 (p + q) (p + r) (q + r))                                                        
 
CT-Area of QG-P8-Triangle in the QL-environment:        (equals 2/9 area QL-Diagonal Triangle)
  • 8 l2 m2 n2 Δ / (9 (l m - l n - m n) (l m + l n - m n) (l m - l n + m n))   

DT-Coordinates QG-P8 in 3 QA-Quadrigons:           (only coordinates of 1st Quadrigon point are given)
  • (2 p2 (p2-q2-r2) : (p2-r2)2 -4 q2 (p2+r2)+3 q4 : 2 r2 (-p2-q2+r2))
 
DT-Coordinates QG-P8 in 3 QL-Quadrigons:           (only coordinates of 1st Quadrigon point are given)
  • (m2 (m2-n2)   :   -2 (l2-m2)(m2-n2)+(-m4+l2 n2)   :   m2 (-l2+m2))
 
DT-Area of QG-P8-Triangle in the QA-environment:
  • S/18                                                                                  (equals 1/9 area QA-Diagonal Triangle)
 
DT-Area of QG-P8-Triangle in the QL-environment:       
  • S/9                                                                                    (equals 2/9 area QL-Diagonal Triangle)
 

Properties:
 
  • QG-P8, QG-P9, QG-P10, QG-P11 are collinear on QG-L5, the 2nd QG-Quasi Euler Line.
  • QG-P8 is the Centroid of the 2nd QG-Quasi Diagonal Triangle: QG-Tr2.
  • QG-P8 is the Midpoint of QG-P1 and QG-P4.
  • QG-P8 is the Reflection of QG-P4 in QA-P1 (QA-Centroid).
  • QG-P8 lies on this line: QA-P26.QL-P12 (-2 : 3) (Eckart Schmidt, July, 2012).
  • The QG-P8 Triangle in the QA-environment and the QA-Diagonal Triangle are homothetic with Homothetic Center QA-P1.
  • The area of the QA-versions of QG-P8 = 1/9 of the area of the QA-Diagonal Triangle.
  • The area of the QL-versions of QG-P8 = 2/9 of the area of the QL-Diagonal Triangle.
  • QA-P26 is the Centroid of the Triangle formed by the 3 QA-versions of QG-P8 (note Eckart Schmidt).
  • QL-P15 is the Centroid of the Triangle formed by the 3 QL-versions of QG-P8 (note Eckart Schmidt).
  • QG-P8 is the Kirikami Center (QG-P15) of the Quadrigon formed by the Centroids of the Component Triangles of the Quadrigon (Eckart Schmidt, August 24, 2012).
  • Let Gi be the Centroids of the Component Triangles Pj.Pk.Pl of the Reference Quadrigon. Now QG-P8 is the Diagonal Crosspoint of the Quadrigon formed by the Midpoints(Pi,Gi) (Eckart Schmidt, September 18, 2012).
  • QG-P8 is the Centroid of the Triangle QG-P1.QA-P25.QA-P26 (all Centroid related points). 

 

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