QL-Ci4: Hervey Circle


Let Hi (i=1,2,3,4) be the Orthocenters of the 4 component triangles of the Quadrangle formed by the Circumcenters of the 4 component triangles of the Reference Quadrilateral.
These 4 Orthocenters lie on one of the 4 defining Lines of the Reference Quadrilateral and are concyclic. The circle through these points is the Hervey Circle.
QL-P3 (Kantor-Hervey Point) is its center.
 
 
QL-Ci4-HerveyCircle-00

Equation/Radius: 
        
Equation in CT-notation:
2 (x + y + z) (AM AN SA x + BL BN SB y + CM CL SC z) - TAL2 (c2 x y +b2 x z + a2 y z) = 0
where:
AM = +a4 m(l-m)(l-n) + b4 l (l-m)(m-n) + c4 l (l-n)(m-n) - b2 c2 (m-n) l (2l-m-n)  +  a2 c2 (l-n)(m2+ln-2mn) - a2 b2 (l-m)(2lm-ln-mn)
AN = -a4 n(l-m)(l-n)   + b4 l (l-m)(m-n) + c4 l (l-n)(m-n)  - b2 c2 (m-n) l (2l-m-n)  -  a2 c2 (l-n)(l m-2ln+mn) - a2 b2 (l-m)(lm-2mn+n2)
BL = -a4 m(l-m)(l-n)  - b4 l(l-m)(m-n)   + c4 m(l-n)(m-n) + b2 c2 (m-n)(l2-2ln+mn) + a2 c2 (l-n) m (l-2m+n) + a2 b2 (l-m)(2lm-ln-mn)
BN = -a4 m(l-m)(l-n) + b4 n(l-m)(m-n) + c4 m(l-n)(m-n) - b2 c2 (m-n)(lm+ln-2mn) + a2 c2 (l-n)m(l-2m+n) + a2 b2 (l-m)(lm-2ln+n2)
CM = +a4 n(l-m)(l-n) + b4 n(l-m)(m-n) + c4 m(l-n)(m-n) - b2 c2 (m-n)(lm+ln-2mn) + a2 c2 (l-n)(2lm-m2-ln)  - a2 b2 (l-m)n(l+m-2n)
CL = +a4 n(l-m)(l-n)  + b4 n(l-m)(m-n) - c4  l (l -n)(m-n) + b2 c2 (m-n)(l2-2lm+mn) + a2 c2 (l-n)(lm-2ln+mn) - a2 b2 (l-m)n(l+m-2n)
TAL = (a - b - c)(a + b - c)(a - b + c)(a + b + c) (l - m)(l - n)(m - n)

Radius2 of Circle in CT-notation:                 
   (a2         (l - m) (l - n) + b2       (m - l) (m - n) + c2       (n - l) (n - m))
* (a2 m n (l - m) (l - n) + b2 l n (m - l) (m - n) + c2 l m (n - l) (n - m))
*   a2 b2 c2   /   (256 Δ4 (l - m)2 (l - n)2 (m - n)2) 
where:  Δ = Area = 1/4 √[(a + b + c) (-a + b + c) (a - b + c) (a + b - c)]

Properties:
  • QL-P3 is the Center of QL-Ci4.
  • The Reflection of QL-P1 in QL-P5 lies on QL-Ci4 (Bernard Keizer, March 28, 2013).
  • The Hervey Circle has the same radius as the Miquel Circle (QL-Ci3).
  • The Hervey Circle is the Reflection of QL-Ci3 in QL-P5.

 

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