QG-P19: Quasi Isogonal Conjugate of QG-P1

 
QG-P19 is the intersection of the reflections of the two QA-DT-side lines unequal QG-L1 in the angle bisectors of the opposite sides of the quadrigon.
This point and its properties are found by Eckart Schmidt (December 12, 2012).
 
 
QG-P19-Quasi Isogonal Conjugate QG-P1-01
                       
 
Coordinates:
CT-Coordinates in 1st QA-Quadrigon:
  • (a2 q r ((p + q) (b2 p + a2 q) r2 + c2 p q (p q - r2))  :  -p r (b4 p (p + q) r (q + r) - (a2 - c2) q2 (a2 (p + q) r - c2 p (q + r)) + b2 q (-a2 (p + q) (p - q - r) r + c2 p (p + q - r) (q + r)))  :  c2 p q (c2 p2 q (q + r) + r (b2 p2 (q + r) + a2 q (-p2 + q r))))
 
DT-Coordinates in 1st QA-Quadrigon:
  • (-(a2 + b2 - c2) p2 (c2 q2 + b2 r2)  :  (b2 p2 + a2 q2) (c2 q2 + b2 r2)  :  (a2 - b2 - c2) (b2 p2 + a2 q2) r2)
 
CT-Coordinates in 1st QL-Quadrigon:
  • (a2 l (-a2 (l - m) (l - n) + c2 (l - n) (-m + n) + b2 (l (m - n) + m (-2 m + n)))  :  b2 m (a2 l2 + c2 n2)  :  c2 n (-a2 (l - m) (l - n) + c2 (l - n) (-m + n) + b2 (l (m - n) + m (-2 m + n))))
 
DT-Coordinates in 1st QL-Quadrigon:
  • (-m2 (a4 (l2 - n2) + b2 (b2 - c2) (m2 - n2) + a2 (c2 (-l2 + n2) + b2 (l2 + m2 + 2 n2)))  :  2 b2 c2 (l2 + m2) n2 + c4 n2 (-l2 + n2) + 2 a2 b2 l2 (m2 + n2) + a4 (l4 - l2 n2) + b4 (m4 - l2 n2)  :  m2 (b4 (l2 - m2) + c4 (l2 - n2) - b2 c2 (2 l2 + m2 + n2) + a2 (b2 (-l2 + m2) + c2 (-l2 + n2))))
 
 
Properties:
Now Si.Xj ^ Sj.Xi = Yk,  Si.Yj ^ Sj.Yi = Xk,  Xi.Yj ^ Xj.Yi = Sk.
  • The Triangles formed by the QA-versions of QG-P1, QG-P18 and QG-P19 are pairwise perspective triangles.
Their centers of perspectivity are collinear.
  • QA-P4 lies on the circle defined by the 3 QA-versions of QG-P19.
  • QG-P19 is concyclic with QG-2P2a, QG-2P2b, QA-P4 (the circle is the QL-Tf1 image of QA-Ci1).
  • QG-P19 lies on QA-Cu7 (as 4th intersection with the circle through QG-2P2a, QG-2P2b, QA-P4).
  • The altitudes of the Triangle formed by the 3 QA-versions of QG-P19 are concurrent in a point on QA-P12.QA-P24.
  • The Polar (see Ref-13, Polar) of QG-P19 wrt any circumscribed conic of the Reference Quadrigon is a line through QG-P18.
  • The Triple Triangle of QG-P19 is perspective with all QA-Component Triangles (see QA-Tr-1 for Desmic Triple Triangles).

 

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