QG-P2 Midpoint 3rd QA-Diagonal

We use this terminology.
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. Coordinates:

• (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))
• (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

• (1 : 0 : 1)
• (0 : 1 : 1)
• (1 : 1 : 0)

• (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:
QA-P1.QG-P12                                        = Newton Line
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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