QG-P9: 2nd QG-Quasi Circumcenter

Let P1.P2.P3.P4 be a Quadrigon and let S be its Diagonal Crosspoint QG-P1.
Let Oi i+1 = Circumcenter of Triangle Pi.Pi+1.S (i = cyclic sequence 1,2,3,4) .
Now is O12.O23.O34.O41 a parallelogram.
Remarkable is that the sides of this parallelogram are perpendicular to the diagonals of the Reference Quadrigon.
QG-P9 is the Center of this parallelogram.

Coordinates:

CT-Coordinates QG-P9 in 3 QA-Quadrigons:           (only coordinates of 1st Quadrigon point are given)
•      (-a2 (a2 - c2) q r + a2 (p2 + r2) SA - (a2 - c2) p q SB - (a2 + c2) p r SC + 8 p (p + q + 2 r) Δ:
b2 (c2 p q + a2 p r + c2 p r + a2 q r) + b2 (p2 + p q + q r + r2) SB  :
-c2 (-a2 + c2) p q - (a2 + c2) p r SA + (a2 - c2) q r SB + c2 (p2 + r2) SC + 8 r (2 p + q + r) Δ2)

CT-Coordinates QG-P9 in 3 QL-Quadrigons:           (only coordinates of 1st Quadrigon point are given)
•       (a2 m (b2 l2 + c2 l2 - 2 b2 l m + b2 m2) n - l m (b2 m2 + a2 n2 + b2 n2) SB + SA (-a2 (l2 m2 + l2 n2 + m2 n2) + 2 l m2 n SB) + 8 m (l m2 - 2 l m n + m2 n + 2 l n2 - m n2) Δ2 :
b2 (a2 + b2 + c2) l m2 n - l m (b2 m2 + a2 n2 + b2 n2) SA - b2 (l2 m2 + l2 n2 + m2 n2) SB - m (b2 l2 + c2 l2 + b2 m2) n SC + 8 l n (2 l m - 2 m2 - l n + 2 m n) Δ2 :
c2 l m (b2 m2 - 2 b2 m n + a2 n2 + b2 n2) - c2 (l2 m2 + l2 n2 + m2 n2) SC + SB (-m (b2 l2 + c2 l2 + b2 m2) n + 2 l m2 n SC- 8 m (l2 m - l m2 - 2 l2 n + 2 l m n - m2 n) Δ2 )

CT-Area of QG-P9-Triangle in the QA-environment:
• (a4 q r (p + q) (p + r) + b4 p r (p + q) (q + r) + c4 p q (p + r) (q + r)  -  2 b2 c2 p q r (q + r) - 2 a2 b2 p q r (p + q) - 2 a2 c2 p q r (p + r) ) / (32 Δ (p + q) (p + r) (q + r) (p + q + r))

CT-Area of QG-P9-Triangle in the QL-environment:
• T1 T2 /(64 Δ (l - m)2 (l - n)2 (m - n)2 (-l m + l n + m n) (l m + l n - m n) (l m - l n + m n))
where:                        T1 = a2 m n (l - m) (l - n) + b2 l n (m - l) (m - n) + c2 l m (n - l) (n - m)
T2 = a2 m n (l - m) (l - n) (-3 l m + l n + m n) (l m - 3 l n + m n)
+ b2 l n (m - l) (m - n) (l m + l n - 3 m n) (-3 l m + l n + m n)
+ c2 l m (n - l) (n - m) (l m + l n - 3 m n) (l m - 3 l n + m n)

DT-Coordinates QG-P9 in 3 QA-Quadrigons:           (only coordinates of 1st Quadrigon point are given)
• ((( Sb+a2) (p2-r2)-Sc q2) (Sa p2+Sb q2+Sc r2)-Sb Sc (p-q+r) (p+q-r) (-p+q+r) (p+q+r) :
(Sa p2+Sb q2+Sc r2) (Sa (p2+q2-r2)+Sc (-p2+q2+r2))-(S2+Sa Sc) (-p+q+r) (p+q-r) (p-q+r) (p+q+r) :
-((c2+ Sb) (p2-r2)+Sa q2) (Sa p2+Sb q2+Sc r2)-Sb Sa (p-q+r) (p+q-r) (-p+q+r) (p+q+r))

DT-Coordinates QG-P9 in 3 QL-Quadrigons:           (only coordinates of 1st Quadrigon point are given)
• (a2 m2 (Sb l2+Sa m2-c2 n2) :  -(S2+ Sb Sc) l2 m2+Sb b2 m4+2 S2 l2 n2-( Sa Sb+S2) m2 n2 :  c2 m2 (-a2 l2+Sc m2+Sb n2))

DT-Area of QG-P9-Triangle in the QA-environment:
• (a2 b2 r4+b2 c2 p4+ c2 a2 q4-2 Sa a2 q2 r2-2 Sb b2 p2 r2-2 c2 Sc p2 q2/ (8 S (p-q-r) (p+q-r) (p-q+r) (p+q+r))

DT-Area of QG-P9-Triangle in the QL-environment:
• ((Sc (l2-m2)2+Sb (l2-n2)2+Sa (m2-n2)2) (Sc (l2-m2)2 n4+Sb m4 (l2-n2)2+Sa l4 (m2-n2)2))  / (8S (l2-m2)2 (l2-n2)2 (m2-n2)2)

Properties:

• QG-P8, QG-P9, QG-P10, QG-P11 are collinear on QG-L5, the 2nd QG-Quasi Euler Line.
• QG-P9 is the Circumcenter of the 2nd QG-Quasi Diagonal Triangle: QG-Tr2.
• QG-P9 is the Reflection of QG-P7 in QA-P1 (QA-Centroid).
• QG-P9 is the Midpoint of QG-P1 and QG-P5.
• QA-P1 (QA-Centroid) lies on the circle defined by the 3 QA-versions of QG-P9.
• QL-P9, QL-P16 as well as QL-P17 lie on the circle defined by the 3 QL-versions of QG-P9. The center of the similar circle with QG-P5 lies also on the QG-P9 circle.
• QG-P9 is the Reflection of QG-P7 in the line QG-P2.QG-P12 (which coincides with the Newton Line QL-L1).
• The area of the QA-versions of QG-P9 = the area of the QA-versions of QG-P7.
• The area of the QL-versions of QG-P9 = the area of the QL-versions of QG-P7.
• Let Oi be the Circumcenters of the Component Triangles Pj.Pk.Pl of the Reference Quadrigon. Now QG-P9 is the Diagonal Crosspoint of the Quadrigon formed by the Midpoints(Pi, Oi) (Eckart Schmidt, September 18, 2012).
• QG-P9 lies on the Perpendicular Bisector of the baseline segment of the Miquel Triangle QA-Tr2 (Eckart Schmidt, October 5, 2012).

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