QG-P1: Diagonal Crosspoint

In a Quadrigon the vertices P1, P2, P3, P4 have a fixed order.
The Diagonal Crosspoint is the intersection point of the 2 lines connecting opposite points.

Four points can be ordened in 6 ways:
P1-P2-P3-P4
P1-P2-P4-P3
P1-P3-P2-P4
P1-P3-P4-P2 = reversed sequence of P1-P2-P4-P3
P1-P4-P2-P3 = reversed sequence of P1-P3-P2-P4
P1-P4-P3-P2 = reversed sequence of P1-P2-P3-P4
When we take into account that some sequences are reversed sequences, only 3 types of Quadrigon-orders remain: P1-P2-P3-P4, P1-P2-P4-P3 and P1-P3-P2-P4.
This means that a system of 4 points - also named (complete) quadrangle - consists of 3 quadrigons: P1-P2-P3-P4, P1-P2-P4-P3 and P1-P3-P2-P4.
These 3 quadrigons in a quadrangle produce different Diagonal Crosspoints as can be seen in next picture. In the same way a system of 4 lines - also named (complete) quadrilateral - consists of 3 quadrigons: L1-L2-L3-L4, L1-L2-L4-L3 and L1-L3-L2-L4.
These 3 quadrigons in a quadrilateral produce different Diagonal Crosspoints as can be seen in next picture. Coordinates:
• (p : q : 0),   (0 : q : r)   and   (p : 0 : r)
• (m n : l n : l m),   (m n : l n : l m)   and   (m n : l n : l m)
CT-Area of QG-P1–Triangle in the QA-environment:
• 2 p q r Δ / ((p + q) (p + r) (q + r))                                                              =  area QA-Diagonal Triangle
CT-Area of QG-P1–Triangle in the QL-environment:
• 4 l2 m2 n2 Δ / ((-l m + l n + m n) (l m + l n - m n) (l m - l n + m n))   =   area QL-Diagonal Triangle

• (0 : 0 : 1),   (0 : 1 : 0)   and   (0 : 0 : 1)
• (0 : 0 : 1),   (0 : 1 : 0)   and   (0 : 0 : 1)
DT-Area of QG-P1–Triangle in the QA-environment:
• S / 2
DT-Area of QG-P1–Triangle in the QL-environment:
• S / 2

Properties:
• The 3 Diagonal Crosspoints in a Quadrangle form the QA-Diagonal Triangle (see paragraph QA-Tr1: QA-Diagonal Triangle).
• The 3 Diagonal Crosspoints in a Quadrilateral form the QL-Diagonal Triangle (see paragraph QL-Tr1: QL-Diagonal Triangle).
• QG-P1 lies on these lines:
QG-P4.QG-P8             (2 : -1 = Reflection of QG-P4 in QG-P8)
QG-P5.QG-P9             (2 : -1 = Reflection of QG-P5 in QG-P9)
QG-P6.QG-P10           (2 : -1 = Reflection of QG-P6 in QG-P10)
QG-P7.QG-P11            (2 : -1 = Reflection of QG-P7 in QG-P11)
QG-P15.QA-P1            (2 : -1 = Reflection of QG-P15 in QA-P1)
QL-P8.QG-P3             (-2 : 3 = QL-AntiComplement of QG-P3)
QA-P10.QG-P2           (-2 : 3 = QA-AntiComplement of QG-P2)

Note: The collinearity of QG-P1, QA-P3 and QL-P1 is discussed at Ref-34, QFG-messages #2987, #2989, #2990, #2992, #3000, #3009, #3011, #3027, #3029 and proven synthetically at Ref-60, page 150 (O ~ QA-P3, R ~ QG-P1, Mr ~ QL-P1).
• QG-P1 is the point with minimal sum of distances to the vertices.
• QG-P1 is the Railway Watcher (see QL-L-1) of QG-L4 and QG-L5, the 1st and 2nd QG-Quasi Euler Lines (note Eckart Schmidt).
• is the Diagonal Crosspoint of the 2nd Quasi Incenter Quadrigon (explanation "2nd Quasi", see QG-P8).
• QG-P1 is the Isogonal Conjugate of QG-P16 (Schmidt Point) wrt the Miquel Triangle (QA-Tr2). (Eckart Schmidt, November 26, 2012)
• The Polar (see Ref-13, Polar) of QG-P1 wrt any inscribed or circumscribed conic of the Reference Quadrigon is the 3rd diagonal QG-L1.
• The Polar (see Ref-13, Polar) of any point on QG-L1 wrt any inscribed or circumscribed conic of the Reference Quadrigon is a line through QG-P1.
• The QA-Orthopole (QA-Tf3) of QG-P1 is a point on the Newton Line QL-L1.
• The QA-Möbius Conjugate (QA-Tf4) of QG-P1 is a point on the line QA-P4.QG-P18.
• The QL-Clawson-Schmidt Conjugate (QL-Tf1) of QG-P1 is QA-P4.
• When the Reference Quadrigon P1.P2.P3.P4 is convex, then Area(QG-P1.P1.P2) * Area(QG-P1.P3.P4) = Area(QG-P1.P2.P3) * Area(QG-P1.P4.P1). See Ref-41, pages 27,28.
• The circle defined by the 3 versions of QG-P1 (QA-Ci1) in a Quadrangle is incident with QA-P2 and QA-P30.
• The Triple Triangle of QG-P1 is perspective with all QA-Component Triangles (see QA-Tr-1 for Desmic Triple Triangles).

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