QG-P2 Midpoint 3rd QA-Diagonal

 
We use this terminology.
A “QA-Quadrigon” is a Quadrigon seen as the Component of a Quadrangle.
A “QL-Quadrigon” is a Quadrigon seen as the Component of a Quadrilateral.
The 3rd Diagonal of a QA-Quadrigon is the same line as the 3rd Diagonal of a QL-Quadrigon. However the Midpoint of a QA-Quadrigon is different from the Midpoint of a QL-Quadrigon.
QG-P2 is the Midpoint of the segment on the 3rd Diagonal of a QA-Quadrigon limited by the intersection points with the 3rd Diagonals of the 2 other Component QA-Quadrigons.
 
Construction:
Let S1 = P1.P2 ^ P3.P4 and S2 = P2.P3 ^ P4.P1.
QG-P2 = the Midpoint of S1 and S2.
 
QG-P2-Midpoint3rdDiagonal-01

Coordinates:
 
CT-Coordinates QG-P2 in 3 QA-Quadrigons:          
  • (p (2 p + q + r) : q (p + r) : r (p + q))
  • (p (q + r) : q (p + 2 q + r) : r (p + q))
  • (p (q + r) : q (p + r) : r (p + q + 2 r))
 
CT-Coordinates QG-P2 in 3 QL-Quadrigons:
  • (m – n : -n : m)
  • (n :   n – l   :   -l)
  • (-m : l :   l – m)
 
CT-Area of QG-P2-Triangle in the QA-environment:         (equals ¼ x area QL-Diagonal Triangle)
  • p q r Δ /(2 (p + q) (p + r) (q + r))
 
CT-Area of QG-P2-Triangle in the QL-environment:         (points are collinear)
  • 0

DT-Coordinates QG-P2 in 3 QA-Quadrigons:          
  • (1 : 0 : 1)
  • (0 : 1 : 1)
  • (1 : 1 : 0)
 
DT-Coordinates QG-P2 in 3 QL-Quadrigons:                      
  • (n2 : 0 : - l2 )
  • (0 : n2 : -m2)
  • (m2 : - l2 : 0)
 
DT-Area of QG-P2-Triangle in the QA-environment:        (equals ¼ area QA-Diagonal Triangle)
  • S / 8
 
DT-Area of QG-P2-Triangle in the QL-environment:         (points are collinear)
  • 0

Properties:
 
  • QG-P2 lies on these lines:
        QG-P1.QA-P10
        QG-P13.QG-P14
        QA-P1.QG-P12                                        = Newton Line
        QA-P5.QG-P4
        QA-P20.QG-P15                                     (1 : 1 => QG-P2 is Midpoint of QA-P20 and QG-P15)
        QG-2P2 a/b.QG-2P3 a/b                     = QG-L1 = 3rd Diagonal Quadrigon
  • QG-P2 is the fourth harmonic point of QG-P12 (Inscribed Harmonic Conic Center) on the Newton Line (QL-L1) wrt the midpoints of the diagonals (note Eckart Schmidt).
  • The triangle formed by the 3 QA-Versions of QG-P2 is the medial triangle of the QA-Diagonal Triangle.
  • The 3 QL-Versions of QG-P2 are 3 points on the Newton Line.
  • The Polar (see Ref-13, Polar) of QG-P2 wrt QG-Co1 as well as QG-Co2 is the line QG-P1.QG-P3.
  • The circle defined by the 3 versions of QG-P2 (QA-Ci2) in a Quadrangle is incident with QA-P17, QA-P29, QA-P36 and the foci of the QA-parabolas (QA-2Co1).
  • The Triple Triangle of QG-P2 is perspective with all QA-Component Triangles (see QA-Tr-1 for Desmic Triple Triangles).

 
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