QA-Cu3: QA-DT-P10 Cubic

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)

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.

MATHEMATICS