QL-Ci2: QL-Medial Circle Diagonal Triangle


QL-Ci2 is the Medial Circle of the QL-Diagonal Triangle (see paragraph QL-Tr1).
QL-P11 is its center.
 
QL-Ci2-QL-MedialCircle-DT

Equations/Radii:
Equation in CT-notation:
    l3 (m - n) (b m - c n)(b m + c n) x2
+ m3 (l - n) (a l - c n) (a l + c n) y2
+ n3 (l - m) (a l - b m) (a l + b m) z2
+ l m (( a2+b2-c2) l2 m2 + (c2-b2) l2 m n + (c2-a2) l m2 n - 2 c2 l m n2 + c2 l n3 + c2 m n3) x y
+ l n (( a2-b2+c2) l2 n2 + (b2-a2) l m n2 + (b2-c2) l2 m n - 2 b2 l m2 n + b2 m3 n+ b2 l m3) x z
+ m n ((-a2+b2+c2) m2 n2 + (a2-c2) l m2 n + (a2-b2) l m n2 - 2 a2 l2 m n + a2 l3 m + a2 l3 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 / (16 Δ2 (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:
            SA x (-x+y+z) + SB y (x-y+z) + SC (x+y-z) z = 0
Radius2 of Circle in DT-notation:  
            a2 b2 c2 / (16 S2)

Properties:
  • These points lie on QL-Ci2:
            QL-P1: Miquel Point = Focus 1st QL-Parabola
            QL-P25: Focus 2nd QL-Parabola
            QG-2P5a/b: Intersection points QG-Diagonals with QG-Ci2
            QG-P3: Midpoint 3rd QL-Diagonal (evident property)
            QG-P17: Projection QG-P1 on QG-L1
  • QL-P11 is the center of the Medial Circle.
  • The 2nd intersection point of QL-Ci2 and QL-Ci6 lies on QL-P8.QL-P17.QL-P25. See Ref-34, Eckart Schmidt, QFG-message #1666.