QG-Tf2: QG-Quasi Isogonal Conjugate


The QG-Quasi Isogonal Conjugate of a point P is the intersection point of the reflected P-cevians in the two angle bisectors of the 2 pairs of opposite sides of the Quadrigon.
This conjugate has been described for points QA-P2 and QA-P4 in Ref-16, page 8.
QG-Tf2-Quasi Isogonal Conjugate-01mut
 
There is another way to construct the QG-Quasi Isogonal Conjugate.
By reflecting the 3rd Diagonal in the Angle Bisectors of the 2 pairs of opposite sides of the Quadrigon and making these reflected lines intersect in point QG-P18.
Now the QG-Quasi Isogonal Conjugate equals the normal Isogonal Conjugate wrt Triangle QG-2P2a.QG-2P2b.QG-P18 (Eckart Schmidt, October 5, 2012).

QG-Tf2-QuasiIsogonalConjugate-20

Special is also that QG-P18 coincides with the Clawson-Schmidt Conjugate of QG-P17, which is the projection of the Diagonal Crosspoint (QG-P1) at the 3rd Diagonal (QG-L1).

Coordinates:
Let P (u : v : w) be a random point to be transformed, then:
 
CT-Coordinates QA-Quasi Isogonal Conjugate in 3 QA-Quadrigons:
  • (-a2 (r v - q w) (b2 p (p + q) r w + a2 q (p + q) r w - c2 p q (-q u + p v + r w))   :
       a4 q2 (p + q) r u w + c4 p q2 (q + r) u w - b2 c2 p q (p + q - r) (q + r) u w - b4 p (p + q) r (q + r) u w + a2 (-b2 q (p + q) r (-p + q + r) u w - c2 q2 (q r u v - p r v2 + p q u w + 2 p r u w + q r u w + p q v w)) :
       c2 (q u - p v) (c2 p q (q + r) u + b2 p r (q + r) u + a2 q r (-p u - r v + q w)) )
  • (-a2 (r v - q w) (c2 p q (p + r) v + a2 q r (p + r) v - b2 p r (-r u + q v + p w)) :
        -b2 (r u - p w) (c2 p q (q + r) u + b2 p r (q + r) u - a2 q r (p u - r v + q w)) :
       -a4 q r2 (p + r) u v - b4 p r2 (q + r) u v + c4 p q (p + r) (q + r) u v + b2 c2 p r (p - q + r) (q + r) u v + a2 (c2 q r (p + r) (-p + q + r) u v + b2 r2 (2 p q u v + p r u v + q r u v + q r u w + p r v w - p q w2)) )
  • (-b4 p2 (p + q) r v w - c4 p2 q (p + r) v w + a4 q (p + q) r (p + r) v w + b2 c2 p2 (-q r u2 + p r u v + p q u w + p q v w + p r v w + 2 q r v w) + a2 (c2 p q (p + q - r) (p + r) v w + b2 p (p + q) r (p - q + r) v w) :
       b2 (r u - p w) (b2 p (p + q) r w + a2 q (p + q) r w + c2 p q (-q u + p v - r w)) :
       c2 (q u - p v) (c2 p q (p + r) v + a2 q r (p + r) v + b2 p r (-r u - q v + p w)) )
CT-Coordinates QL-Quasi Isogonal Conjugate in 3 QL-Quadrigons:
  • (  a2 Tb w    :    b2 (c2 n u + a2 l w) (l u + m v + n w)    :    c2 Tb u )
  • (  a2 Tc v    :    b2 Tc u    :    c2 (b2 m u + a2 l v) (l u + m v + n w) )
  • (-a2 (c2 n v + b2 m w) (l u + m v + n w)    :   b2 Ta w   :    c2 Ta v )
where:    Ta = b2 (l - m) (m - n) u - c2 (l - n) (m - n) u + a2 ( l m u + l n u - m n u + l m v + l n w)
                Tb = a2 (l  - m) (l  - n) v + c2 (l - n) (m - n) v + b2 (-l m u - l m v + l n v - m n v - m n w)
                Tc = a2 (l - m) (l - n) w - b2 (l - m) (m - n) w + c2 (-l n u - m n v + l m w - l n w - m n w)

DT-Coordinates QA-Quasi Isogonal Conjugate in 3 QA-Quadrigons:
  • (-(b2 p2 u+a2 q2 u+2 Sc p2 v)/(2 Sc q2 u+b2 p2 v+a2 q2 v)   :  1  :   -(2 Sa r2 v+c2 q2 w+b2 r2 w)/(c2 q2 v+b2 r2 v+2 Sa q2 w))
  • (1   :   (-a2 q2 v-b2 p2 v-2 Sc q2 u)/(2 Sc p2 v+a2 q2 u+b2 p2 u)   :   (a2 r2 w+c2 p2 w+2 Sb r2 u) / (-2 Sb p2 w-a2 r2 u-c2 p2 u))
  • ((-c2 p2 u-a2 r2 u-2 Sb p2 w)/(2 Sb r2 u+c2 p2 w+a2 r2 w)    :    (c2 q2 v+b2 r2 v+2 Sa q2 w)/(-2 Sa r2 v-c2 q2 w-b2 r2 w)   :  1)
DT-Coordinates QL-Quasi Isogonal Conjugate in 3 QL-Quadrigons:
  • ( 2 (Sc a2 l2+Sc b2 m2+( Sa Sb+S2) n2) (u2 l2+v2 m2-w2 n2)+u v (a4 l4+b4 m4+c4 n4+2 (S2-Sa2) m2 n2+2 (S2-Sb2) l2 n2+2 (S2+3 Sc2) l2 m2)   :
        4 (Sa u-Sb v+Sc w)2 l2 n2-(2 Sc u l2+a2 v l2+b2 v m2+c2 v n2+2 Sa w n2)2   :
        2 ((S2+ Sb Sc) l2+Sa b2 m2+Sa c2 n2) (-u2 l2+v2 m2+w2 n2)+v w (a4 l4+b4 m4+c4 n4+2 (S2+3 Sa2) m2 n2+2 (S2-Sb2) l2 n2+2 (S2-Sc2) l2 m2))
  • ( 4 (Sa u-Sb v-Sc w)2 m2 n2-(2 Sb w n2+c2 u n2+a2 u l2+b2 u m2+2 Sc v m2)2    :
        2 ((S2+ Sa Sb) n2+Sc a2 l2+Sc b2 m2) (-w2 n2+u2 l2+v2 m2)+u v (c4 n4+a4 l4+b4 m4+2 (S2+3 Sc2) l2 m2+2 (S2-Sa2) n2 m2+2 (S2-Sb2) n2 l2)   :
        2 (Sb c2 n2+Sb a2 l2+( Sc Sa+S2) m2) (w2 n2+u2 l2-v2 m2)+w u (c4 n4+a4 l4+b4 m4+2 (S2-Sc2) l2 m2+2 (S2-Sa2) n2 m2+2 (S2+3 Sb2) n2 l2))
  • ( 2 ((S2+ Sc Sa) m2+Sb c2 n2+Sb a2 l2) (-v2 m2+w2 n2+u2 l2)+w u (b4 m4+c4 n4+a4 l4+2 (S2+3 Sb2) n2 l2+2 (S2-Sc2) m2 l2+2 (S2-Sa2) m2 n2)   :
        2 (Sa b2 m2+Sa c2 n2+( Sb Sc+S2) l2) (v2 m2+w2 n2-u2 l2)+v w (b4 m4+c4 n4+a4 l4+2 (S2-Sb2) n2 l2+2 (S2-Sc2) m2 l2+2 (S2+3 Sa2) m2 n2)   :  
        4 (Sc w-Sa u-Sb v)2 l2 m2-(2 Sa v m2+b2 w m2+c2 w n2+a2 w l2+2 Sb u l2)2 )


Properties: 
  • In a Quadrangle QA-P2 and QA-P4 are QG-Quasi Isogonal Conjugated wrt the Component Quadrigons.
  • The pairs (QG-P1,QG-P19), (QG-P15, QG-P16), (QG-P17, QG-P18) are QG-Quasi Isogonal Conjugated (Eckart Schmidt, November/December, 2012).
  • In the component Quadrigons of a Quadrangle the 3 QG-Quasi Isogonal Conjugated points of a point P coincide when P is on the QA-Cu7 Cubic. So QA-Cu7 is self-conjugated wrt QG-Tf2 of the Component Quadrigons.
  • In the component Quadrigons of a Quadrilateral the 3 QG-Quasi Isogonal Conjugated points of a point P coincide when P is on the QL-Cu1 Cubic. So QL-Cu1 is self-conjugated wrt QG-Tf2 of the Component Quadrigons.
  • QG-Tf2 maps every line (not through a vertice) in a conic through the intersection points of the opposite sides of the Quadrigon . Moreover this conic divides every side Pi.Pj of the Quadrigon at a point Sij into two parts Pi.Sij, Sij.Pj with ratio Pi.Sij/Sij.Pj. The product of these 4 side ratios is 1 (note Eckart Schmidt).
  • For any point at QL-Cu1 the QL-Tf1-transformation point (Clawson-Schmidt Conjugate) equals the QG-Tf2-transformation point (Quasi Isogonal Conjugate). (note Eckart Schmidt)


 

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