QA-P9: QA-Miquel Center

 
Derived from the famous Miquel Point occurring in the QL-environment, there also is a Miquel Center in the QA-environment.
It is the common point of the 3 Miquel Circles constructed in the 3 Component Quadrigons of the Reference Quadrangle.
 
 
QA-P9-Miquel-Center-00
Coordinates:                        
1st CT-Coordinate:
a2 T1 T2 (-a2 T3 T4 / (q+r) + b2 T3 T6 / (p+r) + c2 T4 T5 / (p+q))
              - a2 b2 c2 (p+q+r) (-a2 (p+q+r) T7 T8 / (q+r) + T2 T5 T9 / (p+r) + T1 T6 T7 / (p+q))
 
where:
           T1 = a2 q2 + b2 p2 + 2 SC p q                 T2 = a2 r2 + c2 p2 + 2 SB p r
           T3 = b2 r + SA q                                       T4 = c2 q + SA r                                       
           T5 = b2 p + SC q                                       T6 = c2 p + SB r
           T7 = q (SC r - SB q) + p(SA q + b2 r)     T8 = r (SB q - SC r) + p(SA r + c2 q)     T9 = r (S0 q + SA r) + p(-SC r + c2 q)
 
Properties:
  • QA-P9 is the common point of the 3 QA-versions of QL-Ci3 (Miquel Circle).
  • QA-P9 is concyclic with the 3 vertices of the Miquel Triangle (see QA-Tr2).
  • QA-P9 is the Reflection of the intersection point of the QA-Cu1 Cubic and its asymptote in the circumcenter of the Miquel Triangle (note Eckart Schmidt).
  • The 3 mutual intersection points of the 3 construction circles unequal QA-P9 match with the 3 QA-versions of QG-P5. Also they lie on the Nine-point Conic of the Circumcenter Quadrangle of the Reference Quadrangle (this is the Quadrangle formed by the Circumcenters of the 4 component triangles of the Reference Quadrangle).
  • The Reference Quadrangle and the Quadrangle formed by the vertices of the QA-Diagonal Triangle and QA-P4 (Isogonal Center) share the same QA-Miquel Center.
  • The QA-Möbius Conjugate (QA-Tf4) of QA-P9 is QA-P3.
  • QA-P9 is the Miquel Point of the Antipedal Quadrilateral of QA-P4.
  • QA-P9 is the Orthology Center of the QG-P5 Triple Triangle wrt the QL-P4 Triple triangle. See QA-Tr-1.
  • The reference QA and the QA formed by (QA-P4 , QA-Tr1) have the same QA-Miquel Center QA-P9. See Ref-34, Eckart Schmidt, QFG-message #1666.
  • The Simson line of QA-P9 wrt QA-Tr2 is parallel to QA-P3.QA-P4. See Ref-34, Eckart Schmidt, QFG-message #1666.
 

 

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