QG-2P5: Intersection points of QG-Ci2 with the 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.

CT-coordinates QG-5P1a/b in 1^st 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-5P1a/b in 1^st 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-5P1a/b in 1^st 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-5P1a/b in 1^st 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-2Cu1a/b: Perspective Squares Double Cubic
  • QL-Cu1: QL-Quasi Isogonal Cubic

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