QL-Ci1: QL-Circumcircle Diagonal Triangle
QL-Ci1 is the circumcircle of the QL-Diagonal Triangle (see QL-TR1 and QG-P1).
QL-P9 is its center.
Equation of Circle in CT-notation
l2 ( l m + l n – m n) (b2 m2 + c2 n2 – 2 SA m n) x2
+ m2 ( l m – l n + m n) (c2 n2 + a2 l2 – 2 SB l n) y2
+ n2 (-l m + l n + m n) (a2 l2 + b2 m2 – 2 SC l m) z2
+ 2 (SC l2 m2 + SB l2 n2 + SA 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 [34], QFG #357 by Eckart Schmidt.
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