QG-2P5: Intersection points QG-Ci2 with QG-Diagonals

 
QG-2P5a and QG-2P5b are the intersection points of QG-Ci2 with the Diagonals of the Reference Quadrigon..
QG-Ci2 is the circle with the line segment of the side of the QL-Diagonal Triangle opposite the Diagonal Crosspoint QG-P1 as diameter.
 
 
QG-2P5-Intersection points QG-Ci2 with QG-Diagonals-02
 
Coordinates:
CT-coordinates QG-2P5a/b in 1st QA-Quadrigon:
QG-2P5a:        (c2 p3 - a2 p r2  :  c2 p (p - r) (2 q + r) + r (-a2 (p + 2 q) (p - r) + b2 p (p + 2 q + r))  :  -c2 p2 r + a2 r3)
QG-2P5b:        (c2 p3 - a2 p r2  :  p r (b2 (-p + r) + a2 (p + r) - c2 (p + r))  :  c2 p2 r - a2 r3)
 
DT-coordinates QG-2P5a/b in 1st QA-Quadrigon:
QG-2P5a:        (-2 c2 p4 + 2 SB p3 r + 2 SB p2 r2 - 2 a2 p r3  :  c2 p4 - 2 c2 p3 r - (a2 - c2) p2 r2 + 2 a2 p r3 - a2 r4  :  2 c2 p3 r - 2 SB p2 r2 - 2 SB p r3 + 2 a2 r4)
QG-2P5b:        (2 c2 p4 + 2 SB p3 r - 2 SB p2 r2 - 2 a2 p r3  : -c2 p4 - 2 c2 p3 r + (a2 - c2) p2 r2 + 2 a2 p r3 + a2 r4  :  2 c2 p3 r + 2 SB p2 r2 - 2 SB p r3 - 2 a2 r4)
 
CT-coordinates QG-2P5a/b in 1st QL-Quadrigon:
QG-2P5a:        (-m n ((b2 (-l + m) + c2 (l - n)) (m - n) - a2 (-m n + l (m + n)))  :  l n (-b2 m2 + c2 n2) :  l m (b2 m2 - c2 n2)) 
QG-2P5b:        (m (-a2 l2 + b2 m2) n :  l (a2 l2 - b2 m2) n :  -l m (a2 (l - m) (l - n) - b2 (l - m) (m - n) + c2 (l m - l n - m n)))
 
DT-coordinates QG-2P5a/b in 1st QL-Quadrigon:
QG-2P5a:        (0  :  SC  :  SB)
QG-2P5b:        (SB  :  SA  :  0)
 
 
Properties:
  • QG-2P5a and QG-2P5b are collinear with QG-P18
  • QG-2P5a and QG-2P5b lie on these circles:
        QG-Ci2: QL-DT-Thales Circle
        QL-Ci2: Medial Circle QL-Diagonal Triangle
  • QG-2P5a and QG-2P5b lie on these cubics:
        QG-2Cu1 a/b: Perspective Squares Double Cubic
        QL-Cu1: QL-Quasi Isogonal Cubic

 

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