QA-Cu3: QA-DT-P10 Cubic


QA-Cu3 is the locus of the Double Points created by the QA-Line Involution (QA-Tf1) of all lines through QA-P10.
It is a pivotal isocubic of the QA-Diagonal Triangle, invariant wrt the Involutary Conjugate with pivot QA-P10.
QA-Cu3 is a pK(QA-P16,QA-P10) cubic wrt the QA-Diagonal Triangle in the terminology of Bernard Gibert (see Ref-17b). (note Eckart Schmidt)
 
QA-Cu3-QA-DT-P1-P10-Cubic-00
 
Equations: 
Equation CT-notation:
     r2 (p+q) (p+q+2r) (q x - p y) x y
+ q2 (p+r) (p+2q+r) (p z - r x) x z
+ p2 (q+r) (2p+q+r) (r y - q z) y z = 0
Equation DT-notation:
            r2 (x-y) x y + p2 (y-z) y z + q2 (z-x) z x = 0

Properties:
  • The vertices of the Reference Quadrangle and the QA-Diagonal Triangle lie on this cubic.
  • The Midpoints of the sides of the QA-Diagonal Triangle lie on this cubic.
  • The Involutary Conjugate pairs (QA-P1, QA-P20) and (QA-P10, QA-P16) lie on the cubic.
  • The tangents at P1, P2, P3, P4 meet at QA-P10.
  • The tangents at S1, S2, S3 and QA-P10 meet at QA-P16 which is the Involutary Conjugate of QA-P10 on the cubic.


 

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