QA-P41: Involutary Conjugate of QA-P4

 
QA-P41 is the Involutary Conjugate of QA-P4 (Isogonal Center).
It is also a special point because it is the common point of the 3 QA-versions of QG-Ci3 (Quasi Isogonal Circumcenter).
         
 QA-P41-InvolutaryConjugate-of-QA-P4-02
 
 
Coordinates: 
1st CT-Coordinate:
a2 q r (a2 p2 - b2 p2 - c2 p2 + a2 p q - b2 p q + a2 p r - c2 p r + a2 q r)
(-c4 p3 q3 + a2 c2 p3 q2 r - c4 p3 q2 r + b2 c2 p2 q3 r - c4 p2 q3 r + a2 b2 p3 q r2 - b4 p3 q r2 + a4 p2 q2 r2 - b4 p2 q2 r2 + 2 b2 c2 p2 q2 r2 - c4 p2 q2 r2 + a4 p q3 r2 - a2 b2 p q3 r2 - b4 p3 r3 - b4 p2 q r3 + b2 c2 p2 q r3 + a4 p q2 r3 - a2 c2 p q2 r3 + a4 q3 r3)
1st DT-Coordinate:
p2 / (-b2 c2 p4 + b2 c2 p2 q2 - c4 p2 q2 - b4 p2 r2 + b2 c2 p2 r2 + a4 q2 r2)
 
Properties:
  • QA-P41 is the common intersection point of the 3 QA-versions of QG-Ci3 (Quasi Isogonal Circumcircle).
  • QA-P41 lies on the circumcircle of the triangle formed by the 3 QA-versions of QG-P18.
  • QA-P41.QA-P11 // QA-P2.QA-P23
  • QA-P41 lies on the cubics QA-Cu1 and QA-Cu7.
  • The QA-Möbius Conjugate (QA-Tf4) of QA-P41 is the intersection point QA-P2.QA-P4 ^ QA-P3.QA-P32.
  • QA-circumconics intersect QA-Cu1 in two further points collinear with QA-P41. See Ref-34, Eckart Schmidt, QFG-message #1666.
  • The tangent at QA-P4 and the tangents at the vertices of the Diagonal Triangle (QA-Tr1) to QA-Cu1 are concurrent in QA-P41.
  • Let QG-P1a, QG-P1b, QG-P1c be the three QA-versions of QG-P1 and let QG-P18a, QG-P18b, QG-P18c be the three QA-versions of QG-P18QA-P41 is the common point of the circles (QG-P18a,QGP1b,QGP1c), (QG-P18b,QGP1c,QGP1a) and (QG-P18c,QGP1a,QGP1b).