QA-Cu2: QA-DT-P5 Cubic


QA-Cu2 is the locus of the Double Points created by the QA-Line Involution (QA-Tf1) of all lines through QA-P5.
It is a pivotal isocubic of the Diagonal Triangle, invariant wrt the Involutary Conjugate wrt pivot QA-P5.
QA-Cu2 is a pK(QA-P16,QA-P5) cubic wrt the QA-Diagonal Triangle in the terminology of Bernard Gibert (see Ref-17b). (note Eckart Schmidt)
 
QA-Cu2-QA-DT-P1-P5-Cubic-00
 
Equations: 
Equation CT-notation:
   (p+q) (p+q+2r) (q x - p y) x y
+ (p+r) (p+2q+r) (p z - r x) x z
+ (q+r) (2p+q+r) (r y - q z) y z = 0
Equation DT-notation:
   (4 (p4+q2 r2) - (p2+q2+r2)2) (r2 y2-q2 z2) x
+ (4 (q4+p2 r2) - (p2+q2+r2)2) (p2 z2-r2 x2) y
+ (4 (r4+p2 q2) - (p2+q2+r2)2) (q2 x2-p2 y2) z = 0
 
QA-Cu2-QA-DT-P1-P5-Cubic-02
Properties:
  • The lines M12.M34, M13.M24, M14.M23 are the asymptotes of QA-Cu2, where Mij is the Midpoint of Pi.Pj and (i,j) (1,2,3,4).
  • The 3 asymptotes of QA-Cu2 meet at QA-P1.
  • The tangents at P1, P2, P3, P4 meet at QA-P5.
  • The tangents at S1, S2, S3 and QA-P5 meet at QA-P17 which is the Involutary Conjugate of QA-P5 on the cubic.
  • The QA-Cu2 cubic is symmetrical wrt QA-P1 (note Eckart Schmidt).


 

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