QA-Cu5: QA-DT-P1 Cubic
QA-Cu5 is the locus of the Double Points created by the QA-Line Involution (QA-Tf1) of all lines through QA-P1.
It is a pivotal isocubic of the QA-Diagonal Triangle, invariant wrt the Involutary Conjugate with pivot QA-P1.
QA-Cu5 is a pK(QA-P16,QA-P1) cubic wrt the QA-Diagonal Triangle in the terminology of Bernard Gibert (see Ref-17b). (note Eckart Schmidt)
Equations:
Equation CT-notation:
p (2p+q+r)(r y - q z) y z + q (p+2q+r)(p z - r x) x z + r (p+q+2r)(q x - p y) x y = 0
p (2p+q+r)(r y - q z) y z + q (p+2q+r)(p z - r x) x z + r (p+q+2r)(q x - p y) x y = 0
Equation DT-notation:
p2 (-p2+q2+r2)(r2 y2-q2 z2) x + q2 (-p2+q2-r2)(r2 x2-p2 z2)y + r2 (p2+q2-r2) (q2 x2-p2 y2) z = 0
Properties:
- The vertices of the Reference Quadrangle and the QA-Diagonal Triangle lie on this cubic.
- The Involutary Conjugate pair (QA-P1, QA-P20) lie on the cubic.
- The tangents at P1, P2, P3, P4 meet at QA-P1.
- The tangents at S1, S2, S3 and QA-P1 meet at QA-P20 which is the Involutary Conjugate of QA-P1 on the cubic.