QL-Ci1: QL-Circumcircle Diagonal Triangle

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QL-Ci1 is the circumcircle of the QL-Diagonal Triangle (see QL-TR1 and QG-P1).

QL-P9 is its center.

 

Equations/Radii

Equation of Circle in CT-notation:

     l² ( l m + l n - m n) (b² m² + c² n² - 2 S_A m n) x²

+ m² ( l m - l n + m n) (c² n² + a² l²  - 2 S_B l n) y²

+ n² (-l m + l n + m n) (a² l² + b² m² - 2 S_C l m) z²

+ 2 (S_C l² m² + S_B l² n² + S_A m² n²) (l m x y + l n x z + m n y z) = 0

Radius² of Circle in CT-notation:

   (a² l²  - a² l m - b² l m + c² l m + b² m²)

* (a² l²  - a² l n + b² l n - c² l n + c² n²)

* (b² m² + a² m n - b² m n - c² m n + c² n²)

* l² m² n² / (4 Δ ² (l m - l n - m n)² (l m + l n - m n)² (l m - l n + m n)²)

where: Δ = Area = 1/4 √[(a + b + c) (-a + b + c) (a - b + c) (a + b - c)]

Equation of Circle in DT-notation:

            a² y z + b² x z +c² x y = 0

Radius² of Circle in DT-notation:  

            a² b² c² / (4 S²)

Properties:

• QL-P16 (QL-Quasi Circumcenter) and QL-P17 (QL-Adjunct Quasi Circumcenter) and QL-P24 (Intersection QL-P1.QL-P8 ^ QL-P13.QL-P17) lie on QL-Ci1.
• QL-Ci1 is orthogonal wrt the Plücker Circle (QL-Ci5).
• QL-Ci1 is the locus of Euler-Poncelet points (QA-P2) of the Quadrangle formed by the tangential points of the QL-tangential conics. See Ref-34, QFG #357 by Eckart Schmidt.

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