QG-Ci2: QL-DT-Thales Circle

The QL-DT-Thales Circle is the circle with the line segment between the vertices of the QL-Diagonal Triangle (QG-2P3a and QG-2P3b) that are unequal the Diagonal Crosspoint (QG-P1) as diameter.

CT-Equation QG-Ci2 in 1^st QA-Quadrigon:

r (-a² q + b² q + c² q + b² r) x² – (c p – a r) (c p + a r) y² – p (b² p + a² q + b² q – c² q) z²

+ (2 c² p q – a² q r + b² q r – c² q r + a² r² + b² r² – c² r²) x y

+ q (-a² p + b² p + c² p – a² r – b² r + c² r) x z

+ (a² p² – b² p² – c² p² + a² p q – b² p q + c² p q – 2 a² q r) y z = 0

CT-Equation QG-Ci2 in 1^st QL-Quadrigon:

l² (b m – c n) (b m + c n) x² + m² (a² l² – a² l n + b² l n – c² l n + c² n²) y² + n² (b m- a l) (b m + a l) z²

+ l m² (a² l + b² l – c² l – a² n + b² n + c² n) x y + m² n (a² l + b² l – c² l – a² n + b² n + c² n) y z

+ l n (2 b² m² – a² l n + b² l n – c² l n) x z = 0

DT-Equation QG-Ci2 in 1^st QA-Quadrigon:

(c² p² – a² r² + b² r²) x² + (2 c² p² – a² r² + b² r² – c² r²) x y + 2 (c p – a r) (c p + a r) x z +

(a² p² – b² p² + c² p² – 2 a² r²) y z + (-b² p² + c² p² – a² r²) z² = 0

DT-Equation QG-Ci2 in 1^st QL-Quadrigon:

SA x y – SB y² + b² x z + SC y z = 0

Properties

• These points lie on QG-Ci2:
  • QG-2P3a/b (intersection points QG-Ci2 with the 3^rd QG-Diagonal)
  • QG-2P5a/b (intersection points QG-Ci2 with the QG-Diagonals)
• QG-Ci2 is orthogonal wrt circles QG-Ci1, QG-Ci3 and QA-Ci1.
• The intersection points of QG-Ci2 with QG-Ci1 lie on the Perspective Squares Double Cubic QG-2Cu1.
• The intersection points of QG-Ci2 with the diagonals of the Reference Quadrigon (QG-2P5a and QG-2P5b) lie on the cubics QL-Cu1 and QG-2Cu1.

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