QG-P3 Midpoint 3rd QL-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-P3 is the Midpoint of the segment on the 3rd Diagonal of a QL-Quadrigon limited by the intersection points with the 3rd Diagonals of the 2 other Component QL-Quadrigons.

Construction:
Let S1 = P1.P2 ^ P3.P4 and S2 = P2.P3 ^ P4.P1. S1.S2 is the 3rd Diagonal.
Let S3 = P1.P3 ^ S1.S2 and S4 = P2.P4 ^ S1.S2.
QG-P3 = the Midpoint of S3 and S4. Coordinates:

CT-Coordinates QG-P3 in 3 QA-Quadrigons:
• (  p (p + r)  : -q (q + r)  :   r (p - q) )
• (-p (p + q)  :  q (-p + r) :  r (q + r) )
• (  p (q - r)   :  q (p + q)  : -r (p + r) )
CT-Coordinates QG-P3 in 3 QL-Quadrigons:
• (m2 n2 : -n l2 (n - m) : m l2 (n - m))
• (m2 (l - n) n   : l2 n2   : -l m2 (l – n))
• (-m n2 (m - l) : l (m - l) n2  :  l2 m2 )
CT-Area of QG-P3-Triangle in the QA-environment:         (points are collinear)
• 0

CT-Area of QG-P3-Triangle in the QL-environment:         (equals ¼ x area QL-Diagonal Triangle)
• l2 m2 n2 / ((-l m + l n + m n) (l m + l n - m n) (l m - l n + m n))

DT-Coordinates QG-P3 in 3 QA-Quadrigons:
• (0 : -q2 : r2)
• (-p2 : 0 : r2)
• (-p2 : q2 : 0)
DT-Coordinates QG-P3 in 3 QL-Quadrigons:
• (0 : 1 : 1)
• (1 : 0 : 1 )
• (1 : 1 : 0)
DT-Area of QG-P3-Triangle in the QA-environment:        (points are collinear)
• 0

DT-Area of QG-P3-Triangle in the QL-environment:         (equals ¼ x area QL-Diagonal Triangle)
• S / 8

Properties:
• QG-P3 lies on these lines:
QG-2P2a/b.QG-2P3a/b                   = QG-L1 = 3rd Diagonal of the Quadrigon
• The Triangle formed by the 3 QL-Versions of QG-P3 is the medial triangle of the QL-Diagonal Triangle.
• The 3 QA-Versions of QG-P3 are collinear points.
• The three QA-versions of QG-P3 are collinear on the tripolar of QA-P16 wrt QA-Tr1. See Ref-34, Eckart Schmidt, QFG#1263.
• The Polar (see Ref-13, Polar) of QG-P3 wrt QG-Co1 as well as QG-Co2 is the line QG-P1.QG-P2.

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