QA-Cu5: QA-DT-P1 Cubic

QA-Cu5 is the locus of the Doublepoints 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 [17b]). (note Eckart Schmidt)

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

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.

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